PRIME IDEALS IN RINGS OF CONTINUOUS FUNCTIONS By CHAWNE MONIQUE KIMBER '^- -^ '^ : '1' ..A A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1999 This work is dedicated to those women who preceded me and to those who are yet to follow. • ^r;-;> ACKNOWLEDGMENTS First and foremost, I express my wholehearted gratitude to my advisor, Jorge Martinez. In the past few years, his humane guidance has helped me to achieve so very much, in fact, more than I would ever have hoped. I follow his example both in becoming a mathematician and a caring teacher, and in the enjoyment of the finer things in life like wine, cheese, and chocolate. Also, sincere thanks go to my committee members: Richard Crew, for show- ing me some algebra; Alexander Dranishnikov, for teaching me a heap of topology; Scott McCuUough, for introducing me to real analysis (back when we were both much younger); and Mildred Hill-Lubin, for expanding my world-view through lit- erature. . " Cheers and warm hugs to my friends, neighbors, and family, especially to the immediate: Johnnie, Charles, Prances, Maribell, Chinene, Jean, and the inimitable Poopy-girl, Cei. TABLE OF CONTENTS ACKNOWLEDGMENTS iii ABSTRACT v CHAPTERS 1 PRELIMINARIES 1 1.1 History 1 1.2 Lattice-Ordered Groups 3 1.3 /-Rings 7 , 1.4 Rings of Continuous Functions 10 1.5 Approaches 15 2 CHARACTERS 18 2.1 Hahn Groups 18 2.2 Lex Kernels and Ramification 20 2.3 Rank 26 2.4 Rank via Z*-Irreducible Surjections 32 2.5 Prime Character 36 2.6 Filet Character 43 3 GENERALIZED SEMIGROUP RINGS 46 3.1 Specially Multiplicative /-Rings 46 3.2 r-Systems and ^-Systems 52 3.3 /-Systems 58 3.4 Survaluation Ring and n*''-Root Closed Conditions 66 4 RAMIFIED PRIME IDEALS 75 4.1 Ramified Points 75 4.2 Ramified G^-points 79 4.3 Ramification via C-Embedded Subspaces 84 5 nvQUASINORMAL /-RINGS 88 5.1 Definitions 88 5.2 {B, m)-Boundary Conditions 94 5.3 J3X, m-Quasinormal and SV Conditions 107 REFERENCES HO BIOGRAPHICAL SKETCH 113 IV Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PRIME IDEALS IN RINGS OF CONTINUOUS FUNCTIONS Chawne Monique Kimber May 1999 Chairman: Jorge Martinez Major Department: Mathematics Given a completely regular topological space X, we wish to determine the order structure of < Spec(C(X)), C>, the root system of prime ideals of the ring of real-valued continuous functions on X; and vice versa. We present four approaches which give partial solutions to these problems. First, we define three measures on < Spec+(G), C>, the set of prime subgroups of a lattice-ordered group, which determine some arithmetic properties of the group, and vice versa. Second, given any root system, we construct a generalized semigroup ring U which is a commutative semiprime /-ring such that < r(7J),C>, its root system of values, is order-isomorphic to the given root system. Then we characterize those non-isolated G^-points whose corresponding maximal ideal is the sum of the minimal prime ideals it contains. Finally, we characterize those spaces X for which C{X) has the property that the sum of any m minimal prime ideals is a maximal ideal or the entire ring. CHAPTER 1 PRELIMINARIES The focus of this dissertation is the order structure of < Spec(C(X)), C>, the spectrum of prime ideals of the ring C{X) of real-valued continuous functions on a topological space, X. To start, we informally present the history and give motivation for the discussion herein. We then review some essentials about lattice-ordered groups, /-rings, and rings of continuous functions in detail and then formally indicate the manner in which this thesis proceeds. 1.1 History Our history begins with the independent research by Cech and Stone in 1937 (see the papers [Ce] and [St]), in which they describe a compactification 13 X of a topological space X which has the property that every real-valued continuous func- tion on X extends to a continuous function on PX. Further, via PX, they establish correspondences between the topological structure of X and certain algebraic prop- erties of its ring C{X) of real-valued continuous functions under pointwise addition and multiplication. For instance. Stone shows that the maximal ideals of the sub- ring C*{X) of bounded functions are in one-to-one correspondence with points of PX. The map p i-^ M*p = {/ € C*{X) : f{p) = 0} witnesses this correspon- dence and is a homeomorphism of topological spaces when the set of maximal ideals of the ring is endowed with the hull-kernel (Zariski) topology. In particular, this shows that for compact Hausdorff spaces X, Y, we have that X = Y if and only if C{X) ^ C{Y). The next significant result came in 1939 when Gelfand and Kolo- mogoroff proved in [GK] that the maximal ideals of C{X) are exactly those of the 2 form MP = {/ e C{X) : p e d^xZ{f)}, where Z{f) = {p e X : f(p) = 0} and pe/3X. In the 1950's, Gillman, Henriksen, Jerison and Kohls began a formal in- vestigation of topological/algebraic correspondences of this form. The elementary techniques and results are recorded in the text Rings of Continuous Functions, [GJ]. Concerning prime ideals, in [GJ, 14.3c] we learn that the prime ideals in C(X), con- taining a given prime ideal, form a chain and in [GJ, 4J], it is shown that the topology on X is closed under countable intersections (i.e., X is a P -space) if and only if every prime ideal of C{X) is maximal. More generally, in [GJ, 14.25] we find that X has the property that every bounded continuous function on set of the form X \ Z{f), for some / e C{X), extends to a continuous function on X (that is, X is an F -space) if find only if every maximal ideal of C{X) contains a unique minimal prime ideal. Knowing these three facts, we can describe the graph of the prime ideal spectrum of C{X) in each case, where vertices are prime ideals and edges indicate set-inclusion. In the F-space situation, the graph is a disjoint set of strands (one for each point of ^X) with no branching; a P-space yields a graph consisting solely of vertices (one for each point of PX). It is from these topological characterizations of the graphical structure of the spectrum of prime ideals of C{X) that we formulate our questions. Roughly speaking, we wish to know: • Is it possible to determine the order structure (under inclusion) of the prime ideal spectrum of the ring of continuous functions of a given topological space? • Conversely, given a graph, is it possible to construct a topological space such that the given graph, in some sense, determines the structure of the spectrum of its ring of continuous functions? It turns out that both of these axe rather ambitious pursuits and the ques- tions must be refined before we can approach them. Extending the knowledge of properties of the prime ideals of C(X), Kohls published a series of papers ([Kl], [K2], and [K3]) in 1957. In [K2], he addresses the properties of chains of prime ideals o{C{X). First he shows that the quotient ring C{X)/P is totally- ordered for every prime ideal P and concludes that the prime ideals of the quotient ring form a chain. Second, it is demonstrated that if P is nonmaximal, then the chain of prime ideals in C{X)/P contains an 771 -sei (that is, a totally ordered set E such that for every pair of disjoint countable subsets A,B c E such that A < B, there exists ce E such that a < c and c < 6 for every aE A and every b e B). Hence, the chain of prime ideals contains at least 2^^ primes. We may thus immediately reduce the second of our questions to only consider those graphs for which each nontrivial edge passes through an 771 -set of vertices. The facts presented in the next three sections show us that the class of graphs to consider can be further reduced. In this dissertation, we continue to refine the questions and present four perspectives-ranging from the very general to the very specific-which give partial results. In order to properly introduce these approaches, we must recall some facts and constructs which are fundamental to the ensuing investigation. 1.2 Lattice-Ordered Groups Let {L, <) be a partially ordered set. If a, 6 6 L are incomparable, then we write a II 6. L is totally ordered if any two elements are comparable. We say that L is a lattice if any two elements a,b e L have a least upper bound and a greatest lower bound, denoted a V 6 and a Ah, respectively. A lattice L is distributive if a A (6 V c) = (a V 6) A (a V c), and dually for all a,b,ceL. A group {G, +, 0, <) with partial order < is a lattice- ordered group (hence- forth, i-group) if it is a lattice and '\ig<h implies that c-\-g < c+h and g+c < h+c for all c E G. The majority of the groups we consider are abelian, so the additive notation here is for convenience. It is important to note that any ^-group is torsion- free [D, 3.5] and its lattice is distributive [D, 3.17]. A (real) vector lattice is an ^-group G which is also an R-vector space such that rg > for all positive g € G and for all positive r e M. By G"*" we mean the set of elements g eG such that g >0. Each element of G may be written as a difference of elements of G~^ :let g'^ = gWO and g~ = (-g) V 0, then g = g^ - g~. This follows from the fact that g+ A g~ = 0. The absolute value of an element is given by |^| = g-^ + g-. In general, we say that a pair of elements g,hEG axe disjoint ii g Ah = O.We will write g <^ hit ng < h for all n e N. An i-homomorphism is a group homomorphism that also preserves the lattice structure. An i-suhgroup if of an ^-group G is a subgroup which is also a sublattice of (G, <). We call an ^-subgroup convex i{Q<g<h€H implies that g e H. G{S) denotes the convex ^-subgroup of G generated by the set S C G. When S = {g}, we write G{g). In fact, G{g) = {/i G G : 3n 6 N, \h\ < n\g\}. In the special case that G(g) = G, we call g a strong (order) unit. Let ^{G) denote the set of all convex ^-subgroups of G ordered by inclusion. This set is a distributive lattice under the operations of arbitrary intersection and Vie/^» = ^(Uie/^«)' where {Hi}ieT Q ^(G) and / is any indexing set; see [D, 7.10] for details. Let S CG. Then the polar of 5" is given by S^ = {g€G: \g\ A |s| = for all s e S}. If 5 is a singleton, say S = {g}, then we write g-^ for the polar of S. Such a polar is called principal. If p^ = 0, then g is termed a weak (order) unit. Note that for any 5 C G, we have that 5^ e (t(G) and (S^)^^ = S-^. Let <P(G) represent the set of polar subgroups of G ordered under set-inclusion. Then <P(G) C (t{G), but in general this is not as a sublattice. Under the operations of arbitrary intersection. ±, and Vie/ ^» ~ (U.e/ ^«)"'""'') where {Hiji^j C ^(G) and / is any indexing set, we have that ^{G) is a Boolean lattice, by [D, 13.7]. The convex ^-subgroups of greatest interest to us are the prime subgroups of G. These are the subgroups H e C(G) for which any one of the following equivalent conditions is satisfied (see [D, 9.1], [AF, 1.2.10], or [BKW, 2.4.1]): 1. VgAh = then g e H oi h e H. 2. U g,h>0 and gAheH then g e H or h e H. 3. The right cosets of H are totally ordered. 4. The convex ^-subgroups of G containing H form a chain. As suggested by the terminology and the second condition listed above, the concept of a prime subgroup is related to that of a prime ideal in a ring. The difference becomes apparent in considering the final equivalent condition which in- dicates that the prime subgroups form a root system. That is, the graph of the prime subgroups of G, in which nodes indicate prime subgroups and edges repre- sent containment going up, has the property that incomparable elements have no common lower bound. Illustrating that prime ideals differ firom prime subgroups, we note that the zero subgroup is the only prime subgroup of the totally ordered group of integers, Z; whereas, the zero ideal and the ideals which are generated by a prime integer comprise the set of prime ideals of Z. The structure of the graph of the prime subgroups is the subject of our investigation. Note that we use Spec+(G) to denote the set, or spectrum, of all prime subgroups of G and to stand for the associated graph. By Zom's Lemma, minimal prime subgroups exist. Let Min+(G) denote the set of all minimal prime subgroups of G.If P e Min+(G) then by [AF, 1.2.11] we 6 have P = U{g^ : 9 i P}- This implies 0{Q) ^ n{P G Min+(G) : F C Q} is the set \J{g^ -giQ}, for a prime subgroup Q C G, by [BKW, 3.4.12]. A basis for an ^-group G is a maximal pairwise disjoint set {gC^i^i C G"*" such that for each i e /, the set {g G G"*" ■ g < gi) is totally ordered. The following is Conrad's Finite Basis Theorem presented as [D, 46.12] and [C, 2.47]. It will figure in our discussion in the next chapter. Theorem 1.2.1. Let G he an i-group. The following are equivalent: 1. G has a finite basis. 2. Min+(G) is finite. 3. <P(G) is finite. 4- There is a finite upper bound on the number of pairwise disjoint elements of G, 5. There is a finite upper bound on the number of elements of strictly increasing chains of proper polars. Another application of Zorn's Lemma establishes the existence of convex i- subgroups which are maximal with respect to not containing a fixed element g €G. Any such subgroup is generally termed a regular subgroup and specifically called a value of g. The set of all regular subgroups of G is usually represented by r(G). Regular subgroups are prime, by [D, 10.4], and a prime subgroup is precisely a convex ^-subgroup which is an intersection of a chain of regular subgroups, [D, 10.8]. In particular, the minimal prime subgroups of G correspond to the maximal chains in r(G). For these reasons, we call the root system given by r(G) the skeleton of Spec+(G). By convention, we view r(G) as a partially-ordered indexing set F whose elements are denoted by lower case Greek letters and then represent the regular subgroups by V-y for 7 6 r. Topologize Spec+ (G) using the hull-kernel (or Zariski) topology whose open base is given by U{g) = {P 6 Spec^.(G) : ^ ^ P} for all 5 6 G. In this topology, Spec+(G) is Hausdorff if and only if Spec4.(G)=Min+(G) by [CM, 1.4]; on the other hand, in the subspace topology, Min+(G) is always Hausdorff and U{g) n Min+(G) is both open and closed for every g €G. The space Spec^. (G) is compact if and only if G has a strong unit, by [CM, 1.3]; it is demonstrated in [CM, 2.2] that Min+(G) is compact if and only if G is complemented, that is, if and only if for every g €G'^ there is an /i 6 G"*" such that g Ah = and gV his a. weak unit. 1.3 /-Rings Let {R, +,-,<) be a ring whose underlying group is an ^-group and satisfies the relations re < sc and or < cs whenever r < s and c > 0. Such a ring is a lattice- ordered ring (abbreviated i-ring). If an ^-ring R also satisfies caA6 = ac A6 = whenever a A 6 = and c > 0, then R is called an f-ring. The following is found in [BKW, 9.1.2]: Theorem 1.3.1. Let R be an i-ring, then the following are equivalent: 1. R is an f-ring. 2. Every polar in R is an ideal. 3. Every minimal prime subgroup of R is an ideal. It is not difficult to verify that every ^-ring which is ^-isomorphic to a subdi- rect product of totally-ordered rings (with coordinatewise operations) is an /-ring. In [BP], it is shown that the converse of this statement holds when we assume the Axiom of Choice (abbreviated, AC). Since we routinely apply AC, let us formally 8 state that we will work within the axioms of ZFC. Then we may use this equiv- alent definition of an /-ring R in order to obtain this list of arithmetic properties given in [BKW, 9.1.10], for a,b,ceR: 1. If c> then c(o V b) = caW cb and (a V b)c = acybc. 2. If c > then c{a Ab) = caAcb and (a A b)c = acAbc. 3. |a||6| = \ab\. 4. If a A 6 = then ab = 0. ' 5. a^ > 0. An It-ideal of an ^-ring i? is an ideal which is a convex ^-subgroup of R. We call an ^-ideal a prime i-ideal'ii it is also a prime ideal. Let Spec(i?) denote the space of all prime ideals of R in the hull-kernel toplogy. Let Max(i2) and Min(iZ) denote the subspaces of maximal and minimal prime ideals, respectively. By property (4) above, we see that prime ^-ideals of an /-ring are prime subgroups, hence, as in the case of ^-groups, the subset Spec^(i2) of prime ^-ideals forms a root system. Denote the subspaces of maximal and minimal prime ^-ideals by Max/(i2) and Min/(i2), respectively. We call a commutative ring semiprime if it contains no nonzero nilpotent elements. In the case of commutative /-rings, we have [BKW, 9.3.1]: Theorem 1.3.2. Let R be a commutative f-ring, then the following are equivalent: 1. R is semiprime. 2. For any a,be R, we have that \a\ A |6| = if and only if ab = 0. 3. Every polar of R is an (.-ideal which is an intersection of prime ideals. 9 I min{R)=Mm{R). 5. R is i-isomorphic to a subdirect product of totally-ordered integral domains. We say that an ^-ring R with multiplicative identity, 1, has the bounded inversion property if a > 1 implies that a is a multiplicative unit. By [HIJo, 1.1], a commutative /-ring R with 1 has the bounded inversion property if and only if Maxi{R) = Max{R). Let A be any commutative ring. Then, in the hull-kernel topology, Min(A) is a Hausdorff space with a base of clopen sets. U Aisa. semiprime ring, then Spec(.(4) is Hausdorff if and only if Min(>l) =Max(^); this occurs if and only if A is von Neumann regular (br absolutely flat,) i.e., for every a E A there exists b £ A such that a = a^6), see [AM, p. 35]. In [HJ] it is demonstrated that if A is a semiprime /-ring, then Min(>l) is compact if and only if A is complemented (i.e., for every a€ A there exists b € A such that ab = and a + 6 is not a zero-divisor). Max{A) is compact for any commutative ring A with identity, and if A is a commutative /-ring with identity which has the bounded inversion property, then the subspace is Hausdorff, see [HJo, 2.3]. Let >l be a commutative ring with identity and P € Spec(>l). Define O{P) = {a€A:3b^P,ab = 0}. If A is also a semiprime /-ring, then this is the same as the ^-subgroup 0{P) defined in the previous section. Recall that the localization of AaX P is the subring, Ap, of the classical ring of quotients of A/0{P) consisting of the elements whose denomi- nator is not in P/0{P). (For a review of this construction and general facts about localizations, see [AM] or [G]). It is the case that Ap is a local ring whose unique maximal ideal is generated by P/0{P) and there is a one-to-one correspondence between prime ideals of Ap and the prime ideals Q of ^ such that 0{P) C Q C P. 10 Thus, if A is also an /-ring with bounded inversion, then by the root system struc- ture of Spec<(>l), we have that Am = A/0{M) since the quotient ring is already local with unique maximal ideal M/0{M). 1.4 Rings of Continuous Functions Let X be a Hausdorff topological space. X is called completely regular (ox Tychonoff) if for every closed set AC X and x € X\A, there exists a real-valued continuous function on X such that f{x) = 1 and f{A) = {0}. Unless otherwise stated, we assume that ail spaces are completely regular. Let C{X) denote the set of real-valued continuous functions on a space X. Under the operations of pointwise addition and multiplication, C{X) is a semiprime ring. Order the ring via: f < 9 a and only if f{x) < g{x) for all x e A". This ordering gives an /-ring structure such that C{X) has the bounded inversion property. Let C*{X) denote the convex ^-subring of bounded functions. The zeroset of /, is the set Z{f) = {x G A" : /(x) = 0}. The complement, coz{f) = X \ Z{f), is the cozeroset of /. By [GJ, 3.6], a Hausdorff space X is completely regular if and only if its topology is the same as the weak topology generated by C{X). Equivalently, the set of all zerosets, Z{X), is a base for the closed sets of such a space, [GJ, 3.2]. Sets A,B C X axe completely separated if there exists / 6 C{X) such that f{A) = {0} and f{B) = {1}. If for every / 6 C{A) there exists / G C{X) such that /U = /> t^^n we say that A is C- embedded in X. Likewise, A is C* -embedded in X if bounded continuous functions on A extend to bounded continuous functions on X. These embedding properties are characterized by complete separation of particular subsets, as follows: 1. Urysohn Extension Theorem [G J, 1.17]: A C A" is C"-embedded in X if and only if any two completely separated sets in A are completely separated in X. 11 2. A C"-embedded set is C-embedded if and only if it is completely separated from every zeroset disjoint from it, [GJ, 1.18]. Recall that a HausdorfF topological space X is normal if any two disjoint closed sets are separated by disjoint open sets. Assuming this stronger separation axiom, the results listed above give rise to the theorem stated in [GJ, 3D], in which the equivalence of the first two statements is known as Urysohn's Lemma. Theorem 1.4.1. Let X be Hausdorff. The following are equivalent: 1. X is normal. 2. Any two disjoint closed sets of X are completely separated. 3. Every closed set of X is C* -embedded in X. 4- Every closed set of X is C-embedded in X. For many reasons, it is often preferable to work with compact spaces. The Stone-Cech compactification 0X is our compactification of choice, since PX is char- acterized by the property that it is (up to homeomorphism) the unique compact space in which X is dense and C*-embedded. There are at least three different ways to construct /SX, we begin with the one based on ultrafilters, described in detail in Chapter 6 of [GJ], which we now summarize. Let X be a completely regular space, let C be a subset of the power set of X and !F CC. .F is a C- filter if ^ .F, it is closed under finite intersections and if for every F 6 .F, the fact that F C F' e C implies that F' € .F. If .F is a Z{X)- filter, then T is also called a z-filter. A maximal filter is an vltrafUter; similarly, a z-ultrafilter is a maximal z-filter. Let ^X be the set of all z-ultrafilters on X which we index by {A*" : p e px). A closed base for the topology on PX is given by sets of the form Z={pepX:ZE AP}, for Z 6 Z{X). Let p 6 ySX and define M" = {/ e C{X) : p € clpxZif)}. 12 The theorem of Gelfand and Kolmogoroff [GK] is stated simply as: Theorem 1.4.2. For a completely regular space X, the set Max(C(X)) is given by {MP-.pe ^X}. In fact, this result gives rise to a homeomorphism of /3X with Max(C(X)). That is, since the sets Z[M^] = {Z{f) : / e M^} are precisely the 2-ultrafilters on X, by [GJ, 2.5], and Theorem 1.4.2 shows that the map p h-> Z[M^\ is the desired correspondence. If p e X, then we will write Mj, and, in this case, the maximal ideal and corresponding z-ultrafilter are called fixed. Otherwise, a maximal ideal and its corresponding z-ultrafilter is called free. It is evident that X is compact if and only if every maximaJ ideal of C{X) is fixed. Maximal ideals are also classified by the residue field C{X)/M^. Identifying the constant functions with their constant, we see that these fields always contain a copy of R We call a maximal ideal real if the field is exactly R; otherwise, the maximal ideal is called hyper-real. This concept is the basis for considering the Hewitt realcompactification of X. Denoted vX, it is the smallest subspace of PX in which X is dense and such that every maximal ideal of C{vX) is real. In fact, by [GJ, 8.5], vX is the largest subspace of pX in which X is C-embedded. With these facts about the maximal ideals firmly in place, we now proceed to consider the nonmaximal prime ideals. We know that every prime ideal of C{X) is convex, by [GJ, 5.5]; so we deduce that Spec(C(X)) is a root system. In order to understand this root system, we are required to consider the properties of other ideals. For instance, for p e pX, the ideals of the form O^ = 0{M^) = {fe C{X) : dpxZif) is a neighborhood of p} are of paramount interest when examining the prime ideals of C{X). One reason is given in [GJ, 7.15]: 13 Theorem 1.4.3. Every prime ideal P in C{X) contains O^ for a unique p e /3X and AfP is the unique maximal ideal containing P. If O^ is prime, then we call p an F-point. If X has the property that O^ is a prime ideal for every p ^ X, then we call X an F-space. We see that in this case, the graph of Spec(C(X)) consists of a set of strands with no branches. Note [GJ, 14.25]: Theorem 1.4.4. Let X be completely regular. The following are equivalent: 1. X is an F-space. 2. pX is an F-space. 3. The prime ideals contained in any given maximal ideal form a chain. 4- Every cozeroset of X is C* -embedded. 5. Any two disjoint cozerosets of X are completely separated. 6. Every ideal of C{X) is convex. 7. Every finitely generated ideal ofC{X) is principal (i.e., C{X) is Bezout^. A special case of an F-point is when Op = M^ and we call p a P -point if this occurs. Call X a P-space if every point of X is a F-point. In this case, the spectrum of C{X) consists only of vertices. Equivalent definitions of F-space are presented in [GJ, 14.29] and are recorded below. First, recall that an ideal / of C(A') is called a z-ideal if / e / and Z{f) - Z(g) implies that g e I. It is immediate from the definitions that M^ and O^ are z-ideals for all p e pX. Note that not all prime ideals are -j-ideals; however, the following says that this is the case in a von Neumann regular ring. 14 Theorem 1.4.5. Let X be completely regular. The follomng are equivalent: 1. X is an P -space. 2. vX is an P-space. 3. Every prime ideal of C{X) is maximal. 4. Every cozeroset of X is C -embedded. 5. For each f G C{X), the zeroset Z{f) is open. 6. Every ideal ofC{X) is a z-ideal. 7. For every f 6 C{X), there exists g G C{X) such that f = gf^ (that is, C{X) is von Neumann regulaxj. We now recall the definitions of other types of spaces which are useful to us. X is basically disconnected if the closure of any cozeroset is clopen. X is extremally disconnected if any open set has open closure. Discrete spaces are extremally dis- connected; extremally disconnected spaces are basically disconnected and all such spaces are F-spaces by [GJ, 14N.4]. Every P-space is basically disconnected by [GJ, 4K.7]. A space is a quasi-F space if every dense cozeroset is C*-embedded. Clearly, from [GJ, 14.25], we see that every F- space is quasi-F. The converses of the pre- ceding statements do not hold. That is, these are distinct classes of spaces, as we now illustrate. Example 1.4.6. Consider the following spaces: 1. Let W be a free ultrafilter on N. Let E = N U {a}, in which points of N are isolated and neighborhoods of a are of the form U U {a}, where U eU. Then E is an extremally disconnected subspace of ^N, but not a P-space. In particular, O,^ is a prime ideal which is not maximal; see [GJ, 4M]. Therefore, E is an F-space. 1$ 2. Let £) be an uncountable set. Let AD = D U {A}, where points of D are isolated and a neighborhood of A is given by any cocountable set containing it. Then AD is basically disconnected, but not extremally disconnected by [GJ, 4N.3]. Moreover, the topological sura X = AD U E is basically disconnected, but neither extremally disconnected nor a P-space, by [GJ, 4N.4]. 3. The corona, /?N \ N is a quasi-F space which is an F-space, yet not basically disconnected; see [GJ, 6W.3, 140]. 1.5 Approaches : Starting as generally as possible in Chapter 2, we define three cardinal- valued characters on the spectrum of prime subgroups of an ^-group. The value of each measure determines a portion of the arithmetic and/or polar structure of the £- group, and vice versa. For instance, we define the prime character, 7r(G) of an ^-group, G to be the least cardinal k such that for any family {Qa}a<K Q Min+(G), of distinct minimal prime subgroups, we have that Vq<k Qc, is the smallest convex ^-subgroup of G containing all the elements of Min+(G). Roughly speaking, it is a measure of the complexity of minimal paths in the graph of Spec+(G) between minimal prime subgroups. We will show that the measure being finite satisfies the following, where lex(G) denotes the smallest convex ^-subgroup of G containing all the elements of Min+(G) : Proposition 1.5.1. Let G be an i-group and m a positive integer. The following are equivalent: 1. 7r(G) =m < oo. 2. m is minimal with respect to the property that lex(G) = G(Uj" ^ aj^) for any m painvise disjoint positive elements, {aj}^^ C lex(G)+. 16 3. m is minimal such that for any prime P ^ lex((7), the chains of proper polars in P have length at most m — 1. Chapter 3 is devoted to a discussion of the properties of F(A, R), the gener- alized semigroup ring of real- valued maps on a root system A (which has a partially defined associative operation, -I-) each of whose support is the join of finitely many inversely well-ordered sets. The ring structure on this group is introduced in [Cl] and [C2]; we endow this ring with an /-ring structure. In particular, we show that if (A, -I-) is a root system such that each of the following holds: 1. + is associative (when it makes sense); 2. if a, /? G A are comparable, then a -j- /?, /? -f- a are defined; 3. if a < /? and a -I- 7, /? + 7 are defined, then a -I- 7 < /? -f- 7 and if 7 -I- a, 7 -I- /? are defined then 7 -(- a < 7 -I- /?; 4. and if n is maximal, then 6 + fi,fi + 6,fi + fj, are defined and 6 + fj, = ^ + S = 6 for every (J < /x, then F(A,R) is an /-ring if and only if 6 = a + implies a,l3 > 6. And when this occurs, the /-ring is seraiprime and satisfies the bounded inversion property. Moreover, by [CHH, 6.1], given any root system A, one of these /-rings has A order-isomorphic to its root system of values. Thus, the second of our questions is answered in the class of /-rings on the level of skeletons. However, the solution to the second problem remains unclear in the smaller class of rings of continuous functions. To gain a modicum of clarity on the situation, we look to the work of Attilio LeDonne, published in 1977 in [Le], in which he addresses the incidence of branching in the graph of Spec(C(X)). He shows, for instance, that the root system branches at every prime 2;-ideal when X is a metric space. In [Le, §2], LeDonne includes a result of DeMarco which states that there 17 is branching at each Mp when X is a first-countable space and p is non-isolated. In Chapter 4, we show that, for a non-isolated G^-point of a completely regular space, there is branching at Mp if and only if X \p is not C*-embedded in X. This result is then used to examine branching in Spec(C(X)) when X is not necessarily first-countable. Both of our questions are addressed in Chapter 5, in which we generalize a few of the results of Suzanne Larson on quasinormal f -rings that are found in the series of papers [Lai], [La2], and [La3]. The semiprime commutative quasinormal /-rings with identity are the ones having the property that the graph of the root system of prime ^-ideals does not contain a subgraph of the form: A (1.1) By [La3, 3.5], a normal space X has the property that C{X) is quasinormal if and only if d{U) PI d{V) is a P-space for any disjoint cozerosets U,V C X. Our generalizations similarly describe those normal spaces X for which Spec(C(A')) does not contain a subgraph of any of the following forms: k k k k (1.2) where n,k,ai,...an are positive integers satisfying some specified conditions. CHAPTER 2 CHARACTERS We seek a collection of measures on root systems whose values will determine some portion of the structure of a lattice-ordered group. In this chapter we describe three such measures: rank, prime character, and filet character. The rank measures the width of a connected component of the spectrum, the prime character determines, roughly speaking, the complexity of minimal paths between minimal primes, and the filet character counts the maximum length of a chain of branching incidences. The first sections of this chapter are a review of two constructs essential to the discussion to follow. 2.1 Hahn Groups To begin, we recall a method of constructing examples of ^-groups having a specified root system as the skeleton of its prime spectrum. Let A be a root system and define y (A, R) = {u : A ^ R : supp(w) has ACC}, where supp(t;) = {5 G A : v{S) 7^ 0}. V{A, R) is an ^-group under pointwise addition ordered by the relation: u > if and only if v{6) > for every maximal element 6 € supp(i;). This ^-group is called a Hahn group. In the paper of Conrad, Harvey and Holland [CHH], it is demonstrated that any abelian ^-group can be embedded in a Hahn group of a more general description than we give here. Of interest to us is the ^-subgroup of maps with finite support denoted by E(A,R) and the £- subgroup of maps whose support is the join of finitely many inversely well-ordered sets, denoted by F(A,R). Clearly, E(A,R) C F(A,R). 18 19 The proof of the first statement of Proposition 2.1.1 is analogous to that of Theorem 6.1 in [CHH]. This establishes that r(E(A,R)),r(F(A,R)) and A are isomorphic as partially-ordered sets. For the sake of completeness, we present an elementary proof of this fact for the case of F(A,R), although the result is easily obtained from the theory of finite- valued ^-groups. The proof is identical in the case ofE(A,R). Recall that an ^-group is finite-valued if each element has only a finite number of values. A special value is a prime subgroup which is the unique value of an element. An ^-group G is finite-valued if and only if every value of G is special and if and only if every element of G is a finite sum of pairwise disjoint special elements; for details, see [AF, 10.10]. If G has a set S of special values such that 5 is a filter and OS = {0}, then G is called special-valued. Proposition 2.1.1. Let A be a root system. For each 5 € A define Vs = {f G F(A, R) : t;(7) = when 7 > 6}. Each Vg is a special value. Further, every value of an element of F{A, R) is of the form Vs for some 6. Thus, A is the skeleton o/Spec^(F(A,R)) and F(A,R) is finite-valued. ' Proof: Let 6 G A and let xs ^ F{A, R) be the characteristic function on {5}. Then Xs ^ Vg and we will show that Vs is the unique value of xs- Let V be a value of xs and let v G V+ \ V^. Then there exists j >S such that 7 is maximal in supp(u), and, hence, t;(7) > 0. If (J < 7, then < xe < v, a. contradiction. If (J = 7, then there exists a positive integer n such that < x* < ^^ and hence x,? € F by convexity, which is a contradiction. Thus V = Vs. ' ^. Let u € F(A, R)"*" and let V be a value of v. Let D be the finite set of maximal elements of supp(t;). Then the characteristic function xd is not in V; else. 20 there exists an integer n such that <v < nxo, a contradiction. Since V is prime and the set {xs ■ 5 E D] is pairwise disjoint, there exists a unique element 5 € D such that xs t ^- By the above, we know that V C Vs. Finally, since v ^ Vs, we have that V = V^, as desired. The final statement follows from [AF, 10.10] since we have shown that every value is special. ■ 2.2 Lex Kernels and Ramification Throughout, we will describe the location of a prime subgroup in the graph in reference to a designated convex ^-subgroup, called the lex kernel of an ^-group G and denoted by lex(G). It is the least convex ^-subgroup containing all the minimal prime subgroups of G. It is always the case that lex(G) is a prime subgroup [D, 27.2] which is normal in G [D, 27.13]. The following is a summary of a part of the discussion of lex kernels in [D, §27] and gives a description of the ^-subgroup in terms of its generators. Proposition 2.2.1. Let G be an (.-group and let C be a convex i-subgroup. The following are equivalent: 1. C = lex(G). 2. C is the least prime subgroup such that if < g ^ C then g > h for every heC. S. C is the convex i-subgroup of G generated by {g ^G : g \\0}. 4.C = {0}u{g€G: ^gu92,...9n e G,g \\ g^ || 52 || •.• || g^ \\ 0}. 5. C is the convex (-subgroup of G generated by the nonunits of G. 6. C is prime and is the smallest among all convex (-subgroups of G which are comparable with every convex (-subgroup of G. 21 7. C is the maximal convex i-subgroup of G such that lex(C) = C. 8. C is the supremum of the proper polars ofG in the lattice of convex i-subgroups of a. It is natural to now introduce a concept which we will discuss in more detail in Chapter 4. This is a generalization of a concept from [Le]. Let >1 be a commutative ring with identity and for each a € A, let Max(a) = {M € Max{A) : a € M}. Recall that an ideal / of >1 is a z-ideal if a G / and Max(a) = Max(6) imply that 6 6 /. Definition 2.2.2. Let A be a commutative f-ring with identity. A prime i-ideal P is ramified if it is the sum of the minimal prime ideals that it contains. A maximal . ■ ; i j> ■ i-ideal M is totally ramified if every prime z-ideoi contained in M is ramified. A completely ramified ring is one in which every prime z-ideal is ramified. Graphically, a prime ^-ideal P < Ais ramified if and only if it is minimal or if the root system of prime ^-ideals of A branches at P. We begin with the ^-group characterization of ramification. It is the case that a ramified maximal ^-ideaJ M of i4 is the lex kernel of the local /-ring A/0{M). In order to discuss a proper lex kernel in an ^-ring, A, we must operate inside a localization. Henceforth, we will obtain results for local rings and tacitly extend to the general case by referring to localizations. The following characterization of ramified maximal ^-ideals is immediate from Proposition 2.2.1. Corollary 2.2.3. Let Abe a commutative semiprime local f-ring with identity and bounded inversion and let M be the maximal ideal. The following are equivalent: 1. M is ramified. 2. M is the convex i-subgroup of A generated by {f ^ A: f \\Q}. 22 3. M = {0}U{feA: 3h,h,...fn € AJ \\ h II /2 II ••• II k II 0}. 4. M is the convex i-subgroup of A generated by the set {feA:3g€A,g>0,gAf = 0}. 5. M is the convex ^-subgroup generated by the elements of A which are not order units. 6. M is the smallest among all convex £-subgroups of A which are comparable with every convex i -subgroup of A. 7. M is the supremum of the proper polars of A in the lattice of convex i-subgroups of A. It is well-known that the lex kernel of an £-group is a prime subgroup (see [D, 27.2]). We now show that the lex kernel of a commutative local semiprime /-ring with identity is an ideal and then give conditions which guarantee that the lex kernel is a prime ideal. Let ^ be a commutative /-ring with identity. Recall that an ideal / < A is pseudoprime if afc = implies a € / or 6 6 /. An ideal J < A'ls semiprime if a E J whenever a"^ e^ J. A is square-root closed if for any <a e A, there exists <b€ A such that a = Ir^. Let a,b € A, then A is n-convex if whenever < a < 6", there exists «€ >l such that a = 6u. Proposition 2.2 A. Let A be a commutative semiprime local f-ring with identity. Then \ex{A) is a prime subgroup which is a pseudoprime i-ideal. If, in addition, A is square-root closed, then lex(A) is a semiprime £-ideal. Proof: Let / 6 lex (A). Then there exists 5 > such that f A g = 0. U af = then af e lex(A); else, af A g — and we conclude again that af 6 lex(i4). Hence lex(yl) is an ideal. j 23 Let N be the set of nonunits of ^4 and recall that lex(A) = A{N). Let oft = 0. If a or 6 is 0, then there isThen by convexity, we see that a^,a~ 6 M^)- Hence a e A{N) and we have that the lex kernel is pseudoprime. Since any prime ideal is semiprime and the lex kernel is the sum of the minimal prime ideals, the lex kernel is semiprime, if A is also square-root closed; see [HLMW, 2.12(d)]. ■ Corollary 2.2.5. Let A be a commutative semiprime local f-ring with identity and bounded inversion and let M be the maximal i-ideal. M is ramified if and only if \ex{A) is a z -ideal. Proof: Since the maximal ^-ideal is the only 2-ideal of a local /-ring, this is immediate. ■ ., ^ . . . Corollary 2.2.6. If A is a commutative local 2-convex semiprime f-ring with iden- tity which is square-root closed, then the lex kernel of A is a prime i-ideal. an ■ Proof: By the remark after [La4, 4.2], under these hypotheses, we have that ^-ideal is a prime ideal if and only if it is pseudoprime and semiprime For the remainder of this section, let G be an abelian lattice-ordered group. Recall the following for H an ^-subgroup of G. H is rigid in G if for every h e H there is g € G such that /i-'-'- = g^-'-. It is shown in [CM, 2.3] that if H is rigid in G then the contraction of minimal prime subgroups of G to minimal prime subgroups of /f is a homeomorphism of minimal prime spaces. If /f € <t{G), then H is very large in G if it is not contained in any minimal prime subgroup of G. It is shown in [CM] that ii H e €{G) then H is very large in G if and only if H is rigid in G. It turns out that ramification in a rigid subring indicates global ramification and vice versa. This is a direct consequence of the lex kernel correspondence demonstrated below. 24 We will also need the following facts (see [BKW, 2.4.7, 2.5.8]): Proposition 2.2.7. Let H G it{G). 1. The contraction map from the set of prime (.-subgroups ofG not containing H to the set of prime i-suhgroups of H is an order-preserving bijection. 2. IfV is a value ofh€H in G, then V t-^VDH is a bijection between the set of values of h in G and the set of values of h in H. :■ Proposition 2.2.8. Let H < G be a convex i-subgroup. Assume that u € H is a weak unit of H and a weak unit of G. Let V be a value of u in G. Then we have that VnH = \ex{H) if and only ifV = lex(G). Proof: Assume that H C P e Min+(G) then « e P and we have that u-^° ^ P by [AF, 1.2.11]. This is a contradiction since u-'^^ = G P. Thus H is rigid in G since it is a convex ^-subgroup which is very large in G. Assume that V n if is the lex kernel of H. Then V H H is the least convex ^-subgroup of H containing all the minimal prime subgroups of H. Since H is rigid in G, by the bijection given in the first part of Proposition 2.2.7, all the minimal prime subgroups of G are contained in V, and V is the least such convex ^-subgroup of G. That is, if W" ^ y also contains the minimal prime subgroups of G, then WnH is a. convex ^-subgroup of H containing all the minimal prime subgroups of H and hence WnH = VnH. But this says that W f) H e r{H) is a value of u and hence, W Er{G) is a. value of u Therefore V = W. If y is the lex kernel of G then VOH contains all the minimal prime subgroups of H. Thus the lex kernel of H is contained in V H H. Let P C V (1 H he a, prime ^-subgroup of H containing all the minimal prime subgroups of H. Then by Proposition 2.2.7, there exists a prime convex ^-subgroup Q < G not containing H such that P = Q Ci H and since we have a rigid embedding, Q contains all the 25 minimal prime ^-subgroups of G. Hence, Q = V, P = Vr\H and V n H is the lex kernel of H. ■ Let A be a commutative semiprime /-ring with identity and bounded in- version. If M € Max{A) then Am is semiprime with bounded inversion. This is a result of the well-known facts that the ^-homomorphic image of an /-ring with bounded inversion has bounded inversion and that Am — A/0{M); see the proof of [La3, 2.7]. Since we must localize an /-ring in order to have a proper lex kernel, the following allows application of Proposition 2.2.8 to /-rings. Proposition 2.2.9. Let B be a commutative semiprime f-ring with identity and let A be a rigid convex f-subring of B. If M G Max/ (A) is such that M = N Ci A for some N e Max/(B), then Am is a rigid convex f-subring of Bn- Proof: Recall that Am = A/0{M) and Bn ^ B/0{N). Define a map <I>:A^Bn by a t-^ a+0{N). This map is an /-ring homomorphism. We show that the image is convex in Bn and that the kernel is 0{M). For a G A, let < b+0{N) < a+0{N). Then there exists n E 0(iV)+ such that < 6 < a+n. If b-n < then < 6 < n and hence b G 0{N) since 0(N) is convex in B. Thus we may assume that < b—n < a. Then b-n e Aajidb + 0{N) = b-n-i-0{N) elm{(f)). Therefore the image of </) is convex in Bff. The kernel of </> is 0{N) n A. It is easy to show that 0{N) OAC 0{M) since M = N n A. For the reverse inclusion, assume that a € A and a ^ 0{N). Since 0{N) is the intersection of the minimal prime ideals of B contained in N, there exists P GMin(B) such that P C AT and a ^ P. By the rigidity of A in B, PnA eMm(A), and therefore a ^ 0{M). We now have that Ker((^) = 0{N) nA = 0{M) and therefore Am is a convex /-subring of Bn. Since Am contains the identity element of Bn, Am is very large in Bn- For rigidity, we need only recall that very large convex embeddings are rigid, [CM]. ■ 26 Corollary 2.2.10. Let B be a commutative semiprime f-ring with identity. Let A be a rigid convex f-subring of B. Let M 6 Maxi{A) be such that M = N n A for some N G Max/(B). Then M is ramified in A if and only if N is ramified in B. Let A* denote the /-subring of bounded elements of the commutative semiprime /-ring A with identity. Note that A* is convex and rigid in A. In his dissertation [Wo], Woodward proves the following fact: Theorem 2.2.11. Let A be a semiprime f-ring with identity and bounded inver- sion. Let M be a mojcimal ideal of A and let M be the unique vaiue of A* containing MnA*. The map M i-¥ M gives a homeomorphism between Max(^) and Max{A*). That is, Max(A*) is the subspace consisting of values of\ in A*. In particular, if M is a real maximal ideal of A, then M HA* e Max(^*). Corollary 2.2.12. Let A be a commutative semiprime f-ring with identity satisfy- ing the bounded inversion property. Let M e Max (A) be real. Then M is ramified if and only if M = M CiA* is ramified in A*. 2.3 Rank The first character on Spec+ (G) that we consider is simply one which counts the minimal prime subgroups contained in a convex ^-subgroup. Definition 2.3.1. The rank, rkG(H) of a convex i-subgroup H < G is the cardi- nality of the set of minimal prime subgroups of G contained in H. If that cardinal is not finite, then we will say that H has infinite rank; we may choose to specify the cardinal when its value is of significance in a discussion. If H is a minimal prime subgroup ofG, then we define rkotH) = 0. This is a variation of the following definition given in [HLMW]: Let ^ be a commutative /-ring with identity and M a maximal ^-ideal of A. The rank of M, denoted rkyi(M), is the cardinality of the subspace of minimal prime ideals of A 27 contained in M. By convention, if the rank of M is infinite and we don't necessarily care about the exact cardinality, we write rk^(M) = oo. The rank of a point p E X, rkxip), is the rank of Mp. The rank of the /-ring A is the supremum of the ranks of the maximal ^-ideals of A, when it exists; the rank of a space X is the rank of c{xy We begin with illustrations of the extremal values of ranks. An ^-group is semiprojectable if for any ^,/i e G+, (5 A h)-^ = G{g-^ U b-^). In [BKW, 7.5.1], it is proved that G is semiprojectable if and only if each prime subgroup contains a unique minimal prime subgroup, which is equivalent to rkcCP) < 1 for every P € Spec^(G). Thus, it is evident that a space X is an F-space if and only if C{X) is semiprojectable which is equivalent to rk(C(A')) < 1. In particular, X is a P-space if and only if C{X) is von Neumann regular, which is equivalent to rk(C(X)) = 0. The one-point compactification of the natural numbers, aN, is an example of a space for which C{X) has infinite rank, [GJ, 14G]. In fact, if a is the point at infinity, then the maximal ideal corresponding to a contains 2* minimal prime ideals - one for each free ultrafilter on N. Moreover, by [HJ, 4.8], this subspace of minimal prime ideals is homeomorphic to the corona, ^OT^ \ N. Proposition 2.3.3 describes a general situation in which we have infinite rank. We recall some definitions. Prom [LZ, 39.1]: let G be a vector lattice, v e G+, and let {^n}^i C G be a sequence. We say that the sequence converges relatively uniformly to g eO along the r^ulator v, and write gn -^ g, if for every e > there exists iVe > such that for all n>Ne,we have that \g-gn\<£v. The sequence is relatively uniformly Cauchy with respect to v if for every e > there exists Ng > such that for all n,m > N^, we have that \gm - gn\ < ev. G is called uniformly complete if for every u € G+, every sequence which is relatively uniformly Cauchy with respect to v relatively uniformly converges along the regulator v. 28 Lemma 2.3.2. Let G be a uniformly complete vector lattice with weak order unit u € G"*". For any set {gj}j&w, */iere exists g eG such that g = fljea;^/- Proof: Let5 = E]^i2-^(b|A«). ■ Proposition 2.3.3. Let G bea uniformly complete complemented vector lattice with weak order unit u e G'^. If for some Q G Spec+(G) we have TkaiQ) > w, then Q contains at least 2* minimal prime subgroups. -,,, . , -^ Proof: Note that, by [CM, 2.2], Mm^{G) is compact since G is complemented. Let V = {P„}neN be a countably infinite set of minimal prime subgroups which are contained in Q which is discrete in the huU-kerael topology on Min+(G). We first show that P is C*-embedded in Min+(G) and conclude that the minimal prime space contains an homeomorphic copy of ;9N. Then we describe the elements of Min+(G) that correspond to the points in this copy of /3N \ N. Let A,B C V he completely separated in V and index them by 7, J C N as A = {Ai : i € 1} md B = {Bj : j e J}. Fix i G / and let Ai e A. For each Bj e B, Let Oi. G A+ \ Bj and &j. G Bf \ Ai. Then A< G OjeJ^i^Ji) =^ ^< ^^ B C \J.^j U{ai.) = t/(Ej6j 2--'(oi. A u)) =^ Li. Then Ki, Li are disjoint closed sets in Min4.(G). For each i G /, generate the disjoint pair ATj, Lj. Then A C d{Ui^iKi) = K def and B C Djg/Lj = L. By Urysohn's Lemma, the disjoint closed sets K^L are completely separated in Min^.(G) since Min+(G) is normal. Consequently, A, B are completely separated in Min+(G) and therefore P is G*-embedded in Min+(G) by the Urysohn Extension Theorem. Finally, by [GJ, 6.5], the closed subset of Min+(G) of minimal prime subgroups in Q contains an homeomorphic copy of /?N. Let W be a free ultrafilter on N. For g £G, let N{g) = {n: g e Pn}. Define a new prime subgroup F by ^ G F if and only if N{g) G U. We show that F is a 29 minimal prime subgroup. The following proof is the same as that for [HJ, 4.8] and for [HLMW, 4.1]. Let g,h e P. P is a. subgroup since N{g - h) D N{g) n N{h) 6 U implies g - h e Phy filter properties. By convexity and since the F^ are prime subgroups, N{g\/h) D N{g) 6 U and N{gAh) D N{g) €U,vfe have that F is a sublattice of G. Thus F is an ^-subgroup of G. Let < 5 < /i G F. Then N{g) D N{h) G U since each F„ is convex, and thus F is convex. Let gAh€ P, then N{g)\JN{h) D N{gAh) e U implies that N{g) or N{h) is in U since U. Thus F is a prime subgroup of G. Let g E P. Since F„ is a minimal prime subgroup for each n, we have that for each n G N{g), there exists an /i„ e G \ F„ such that 5 A /i„ = 0. By Lemma 2.3.2, there exists h e G such that h-^ = rinejvffl) ^n- Then g Ah = and /i ^ F„ for all n e Ar(fli) since h-^ C h^ C F„ for each n G iV(5). Thus iV(5) n H{g) ^ W and hence h ^ P and F is a minimal prime subgroup of G. ■ Recall that a space X is cozero-complemented if for any cozeroset U C. X there exists a cozeroset V C X such that U C\V = and C/ U V is dense in X. A concrete example of a maximal ideal of infinite rank is found in C{X) where X is cozero-complemented and first countable. DeMarco shows in [Le, §2] that rk (Mp) > 2, for any nonisolated point of a first countable space (the result actually says more than this, and we will discuss this in Chapter 4). By modifying DeMarco 's proof, we show that Mp contains infinitely many minimal prime ideals and hence has rank at least 2'. Proposition 2.3.4. Let X he first countable and let p € X he nonisolated. For every m G N there exists a family of m distinct prime ideals which sum to Mp. Moreover, if X is also cozero-complemented, then there exist at least 2* minimal prime ideals contained in Mp. 30 Proof: Since p has a countable base, there exists g e C{Xy such that Z{g) = {p}. Let {Vi}J^i be a neighborhood base at p and define a sequence of real numbers {a„} recursively as follows: for each i, let aj G 9{Vi) such that < • • • < 03 < 02 < oi and lim„_+oo On = 0- Let {x„} C X be a sequence of distinct preimages under g such that Xj € Vj. Then the sequence {x„} converges to p and may be considered as a discrete set in X. Let m > 1 be given and let Wi, W2, . . ,Km be distinct free ultrafilters on the sequence {x„}. Define Pi = {f e C{X) .3AGUi,AC Z{f)}, for each i. DeMarco shows that each of these sets is a prime -s-ideal of C(A") and that Mp is the sum of any two. Thus these prime ideals are noncomparable. We will demonstrate, as DeMarco has done for m = 2, that Mp is the sum of these m noncomparable prime ideals. Let {Aj}J^i be a collection of m pairwise disjoint subsets of {x„} such that Ai eUi, for each 1 < i < m. If Bj = g{Ai), then Bi U {0} is a closed subset of R Thus Bi U {0} is a zero-set of R Choose cpi e C(R) for each 1 < i < m such that Z(^i) = 5i U {0} and Ejli V'i = 1r- Let «i = (fig. Then A^ C Z{g) = Z{ui), hence Uj G Pj. Finally, we have g = Ui+U2 + f- «„. If /i G Mp, then Z{h- + g) = {p} and Z{h+ + g) = {p} By the above, h+ + g,h-+ge YT=i Pi and hence /i+, h-,h € YlT=i Pi- The final statement of the proposition follow from the previous proposition, since the cardinality of y^N is 2*, by [GJ, 9.2]. ■ Now that we have illustrated the extreme cases of 0,1 and infinite rank, we present a result of [HLMW, 3.1], which gives a test for finite rank of a point of a compact space. 31 Proposition 2.3.5. Let X be a compact space. Then p e X has rank n < oo if and only if there exist n pairwise disjoint cozerosets {Uj}^^^ with p 6 0^=1 ^(^i)' and no larger family of pairmse disjoint cozerosets has this property. An /-ring A is called an SV-ring if A/P is a valuation domain (i.e. principal ideals are totally-ordered) for every prime ideal P. A space X is an SV-space if C{X) is an SV-ring. We will discuss this class of rings in more detail in Chapter 3. However, using the above, it is shown in [HLMW, 4.1] that any compact SV-space has finite rank. The validity of the converse of this result is unknown. Presently, our objective is to show that the result in Proposition 2.3.5 does not hold for infinite rank. To demonstrate this, we define a cardinal function on compact spaces and compare its value with a known cardinal invariant. Definition 2.3.6. Let k be a cardinal and let X be a compact space withp € X. Let {Ua)a<K be a family of pairwise disjoint cozerosets of X and call the set na</c <^(^a) a K-boundary. Define p{p, X) to be the infimum over all (infinite) cardinals k such that p is not contained in a K-boundary and let p{X) be the supremum over all the points p €: X of the cardinals p{p, X). Recall that the cellularity of a space X, denoted c{X) is the infimum over all (infinite) cardinals k such that every family of pairwise disjoint open sets of X contains at most k many sets. Proposition 2.3.7. Let X be compact. Then p{X) < c(A')+. Proof: If p(X) > c(X)'*", then there exists a /c-boundary of cardinality greater than the cellularity of the space, which is nonsense. ■ Example 2.3.8. Let r > \!<q. The product space, 2'', where 2 is the two-point discrete space is called the Cantor space of weight r. We show in Example 4.3.4 32 that every point of 2"^ has infinite rank, thus rk(C(2'')) > 2* by Proposition 2.3.3. By [E, 3.12.12(a)], we have that c+(2^) = i^j. Thus, Proposition 2.3.7 gives us that Ko < p(2^) < i^i. Therefore, since Ki ^ 2*, we see that /9(2^) / rk(C(2^)). D 2.4 Rank via Z^-Irreducible Surjections We must first recall a few definitions from [Ha] and [HVW2]. Let X,Y be Tychonoff Hausdorff spaces. Let / : F — > X be a surjective continuous map. Then / is perfect if it is a closed map such that the inverse image of any point is compact. A perfect map is irreducible if proper closed sets of Y map to proper closed sets of X. The pair (F, /) is a cover of X if / is a perfect irreducible surjection from F to X. Let (Fi,/i) and (F2,/2) be covers of X. We define (Fi,/i) ~ (Fa, /a) if there exists a homeomorphism g : Yi -^ Y2 such that /ay = /i. Order the set of ^-equivalence classes of covers via: (Fi,/i) < (Fa, /a) if and only if there exists a continuous map </ : Fi -> Fa such that f^g = /i. A class of spaces C is a covering class if for any space X there exists a least cover (F, /) of X such that Y ^ C. The minimal extremally disconnected, basically disconnected and quasi-F covers of compact spaces are described in [PW], [V], and, respectively, in [DHH], [HudP], and [HVWlj. Certain covering maps allow us to compute the rank of a space externally. A perfect irreducible surjection ^ : F ^ X is Z* -irreducible if for each cozeroset C/ C F, there is a cozeroset V CX such that dviU) = c/y(0~^(F)). This condition on maps is also known as sequential irreducibility and oji-irreducibility. It turns out that a map (f> is Z*-irreducible if and only if C(^) is a rigid embedding of C(X) inside C{Y), by [HaM, 2.2]. Hence we have a homeomorphism Min(C(F)) ^ Min(C(F)) via contraction, by [CM, 2.3]. It is therefore not surprising that these maps are useful for calculating rank. 33 Example 2.4.1. Let X be a compact space. The quasi-F cover of X, {QFX,<l)x), constructed in [HVWl] has the property that (f>x is Z*-irreducible. We summarize this construction. Let Z*{X) = {clxintxiZ) : Z G Z{X)}. For A 6 Z*{X) denote the set of ultrafilters on Z*{X) containing A by A. The authors of [HVWl] show that T{Z*{X)), the compact space of ultrafilters on Z*{X) whose topology has a closed base given hy {A : A e Z*{X)},is quasi-F and define a perfect irreducible surjection (f>x : T{Z*{X)) -^Xhyae T{Z*(X)) maps to the unique point in n{A :Aea}. Then QFX = T{Z*{X)) with the map (^x is the quasi-F cover of X. The map 0x is Z*-irreducible: It is shown in [HVWl, 2.9] that if we have A e Z*{QFX), then <l)x{A) € Z*{X), which is equivalent to the property of Z*- irreducibility. In fact, the quasi-F cover of X is characterized up to equivalence in [HVWl, 2.13] as the only cover {Y,f) of X for which Y is quasi-F and / is Z*-irreducible. D Before we continue, we discuss the question (now answered) which was our motivation for considering this line of investigation. Let A^ be a compact space of finite rank and W = {x e X : Tkx{x) > 1}. In [La2], Suzanne Larson asks if W is always closed in X. The answer is no. Her counterexample, presented at ORD98 (a conference on ^-groups held in Gainesville, FL in 1998) follows: Example 2.4.2. Let W be a free ultrafilter on N. Let E = N U {a} where points of N are isolated and neighborhoods of a are of the form U U {a}, where U eU. Let Ej = E for j = 1,2 and define F = (Ei U E2)/(<7i ~ ^2). Let Yr = Y for each r € R, and let X = (JJ^gR Yr) U {00} where neighborhoods of 00 contain all but countably many copies of the Yr. Then 00 is a P-point which is in the closure of the setW = {xeX : ikxix) > 1}. D 34 We provide a characterization of points of finite rank of a compact space X via Z*-irreducible maps onto X. Let B be an /-subring of A and let 9 denote the natural surjection Max(A) -> Max(J3). The following is proved in [HaM, 2.5]. Lemma 2.4.3. Let A and B be commutative f -rings with identity and hounded inversion. Let B he an f-suhring of A. Then if B is rigid in A, we have that 0{eM) = n{0{N) nB .ON = OM}, and if 9Ni = OM (for j= 1,2) with Ny \\ N2, then {0{Ni) n B) \\ (©(iVa) n B). Let X and Y be compact spaces and (f) -.Y ^ X a. Z*-iiTeducible map. Then e : Max(C(y)) ^ Max{C{X)) is given by M »-> {/ € CiX) : f(l> e M}. Lemma 2.4.4. Let X and Y be compact and (f) : Y -^ X a Z* -irreducible map. Let p G X then 0{Mg) = Mp if and only if <j>{q) = p. Therefore, we have that Op = n{0, n (7(X) : g e r Hp}}- Proof: Let q e 4>~^{p}. If / € ^{^q) then /(^ 6 M, and therefore we have that = /^(9) = /(p) and / e Mp. Let g G Mp-, then {g(f>)iq) = gip) = 0, and hence g<j> e 0{Mg). Thus 6{Mg) = Mp. Conversely, assume ^(g) = r ^ p. By complete regularity, there exists / € C{X) such that f{p) = and /(r) = 1. We have / 6 Mp, but /0(g) = /(r) = 1^0 and hence / ^ 6{Mg). The final statement then follows firom the first and Lemma 2.4.3. ■ Proposition 2.4.5. LetX andY he compact spaces and<f> -.Y -^ X Z* -irreducible. 1. Ifp e X such that rkxip) = n, then \(f>~^{p}\ = n. 2. IfYis an F-space and \(j>~^{p)\ = n then rkx(p) = n. S. IfY has finite rank and \(f>~^{p}\ = n, then rVxip) < 00. Explicitly, we have that rkx(p) = EILi rky(ft) where (j>-^{p} = {ft}?^i. 35 4- IfpGX is an F-point of X, then q e (f>~^{p} is an F-point ofY. Proof: (1) Let <f>~^{p} = {qi}iei for some index set /. For each i e /, choose Qi e Mm(C(F)) such that O,, C Qi. Then by Lemma 2.4.4, it follows that we have Op = HieiiOg, D C{X)) C Qj n C{X) C Mp, for every i e /. By the bijection described in Proposition 2.2.7, the set of minimal prime ideals contained in Mp is given by {Qi n C{X)}i^i and |/| = n. (2) Let <f>~^{p} = {qi}u=i- If 5^ is an F-space, then Og^ is prime for each i and we have that Op = n;Li(0,. n C{X)) C O,. n C{X) C Mp for each i. Thus, by the bijection described in Proposition 2.2.7, the subspace of minimal prime ideals contained in Mp is {Og, n C(X)};Li and xkxip) = n. (3) Let ^~^{p} = {9j}iLi and let the subspace of minimal prime ideals in M, be given by {Qi^ '■ l < j < rkY{qi)}^=,i- Then as above, the subspace of minimal prime ideals contained in Mp is given by {Qi. D C{X) :l <j < rky(9i)}"=i and hence we see that rkx(p) = Yl^=i ^^viQi) < oo, as desired. (4) Let p G X be an F-point and <^~Hp} = {q}- Then Op = O, n C{X) C Mp is a minimal prime. There exists a unique Q e Min(C(F)) such that Op = Qr\ C{X). Since O, n C{X) = Qn C{X) and Q is unique, Og = Qe Min(C(y)). ■ If K = QFX in Proposition 2.4.5, then the third statement is a partial converse of [HLMW, 5.1] which states that if X is compact and has finite rank then QFX has finite rank. The final statement is an extension of [HVWl, 3.12] in which it is shown that the preimage of a P-point of X is a P-point of QFX. The second statement says that if QFX is an F-space, then the points of X of rank one are precisely the points with unique preimage under the covering map ^x- In this light, one should ask when a quasi-F cover of a space is an F-space. Recall that a space X is fraction dense if the classical ring of quotients of C{X) is rigid in the maximal (Utumi) ring of quotients of C(A'). In [HVWl, 2.16], 36 it is demonstrated that QFX is basically disconnected if and only if X is cozero- complemented. In fact, the basically disconnected cover is the quasi-F cover in this case, see [HaM, 2.6.2]. The fact that QFX is realized by the extremally disconnected cover if and only if X is fraction dense is proved in [HaM, 2.4]. By [HudP, 6.2], QFX is an F-space if and only if for any two disjoint cozero-sets Ci, Ca C X, there exist Zi, Z2 e Z{X) such that d C Zi for i=l,2 and int{Zi n Z2) = 0. We now provide an example of a space X such that QFX is an F-space which is not basically disconnected. Recall that a space is a-compact if it is a countable union of compact spaces. Lemma 2.4.6. Let X be a noncompact a-compact locally compact F-space which is not basically disconnected. Let Xi = X2 = PX and define Y be the quotient space of the topological sum of Xi and X2 where pairs of corresponding points of Xj \ X (j=l,2) are collapsed to a single point. Then Y is not quasi-F and QFY is an F-space which is not basically disconnected. Proof: The disjoint union U = X II X is a. dense cozero set of Y which is not C*-embedded in Y. Thus Y is not quasi-F. The quasi-F cover of Y is Xill X2, which is an F-space but not basically disconnected. ■ Example 2.4.7. Let X be the disjoint union of a countable number of copies of the corona jSN \ N. Then X is a a-compact noncompact F-space which is not basically disconnected. Construct Y as defined in Lemma 2.4.6. Then QFY is an F-space which is not basically disconnected. D 2.5 Prime Character The second character we consider counts the minimum number of minimal prime subgroups that we must sum in order to obtain the lex kernel of an ^-group. 37 Definition 2.5.1. Let S be a family of minimal prime subgroups of G. We call S ample ifVS = lex(G). The prime character ofG, denoted '^{G), is the least cardinal so that any family of distinct minimal prime subgroups of that cardinality is ample. Note that i/lex(G) = 0, i.e., if G is a totally-ordered group, then we say n{G) = 1. A prime subgroup properly contained in lex(G) is called embedded. Proposition 2.5.2. Let G be an i-group and m a positive integer. The following are equivalent: 1. 7r(G) = m < oo. g. m = 1 + sup{rk(P) : P € Spec+(G) is embedded}. 3. m is minimal with respect to the property that lex(G) = G(U]^^aj'-) for any m pairwise disjoint positive elements, {aj}^i C lex(G)"'". 4- m is minimal such that for any embedded prime P, the chains of proper polars in P have length at most m—1. Proof: (1) =^ (2) : Let P be embedded. If P contains the m minimal prime subgroups {Qj}]Li C Min+(G) then V^^Qj C P C lex(G). Hence 7r(G) > m. Thus we have shown that 7r(G) < m implies rk(F) < m. Thus by (1), rk(P) < m - 1 and sup{rk(P) : P embedded} < m - 1. If sup{rk(P) : P embedded} < m - 1, then for any family 5 of m - 1 minimal prime subgroups of G, yS is not embedded since rk(v5) > m- 1. Thus V^ = lex(G) and 7r(G) < m- 1. Thus ir{G) = m implies that sup{rk(P) : P embedded} > m - 1. Therefore sup{rk(P) : P embedded} = m + 1. (2) =j> (3) : Let {oj}^! C lex(G)+ be pairwise disjoint. Let g G lex(G) \ GdJjt^ af ) and let V be a value of g such that G(U!^i af) CV C lex(G). Then by the polar characterization of 0(y), [BKW, 3.4.12], we have that Oj ^ 0{V) for each j such that 1 < i < m. By (2), V contains at most m - 1 minimal prime subgroups, {Qi}7^i^- Since 0{V) = fll^"^ Qi, we have by the pigeonhole principle that there 38 must be a minimal prime subgroup, Q, contained in V which fails to contain two of the elements of {aj}f^i. But since these elements are pairwise disjoint, this contrar diets the fact that Q is prime. Therefore, lex(G) = G(Uj° ^ a^). For the minimality of m, let P be an embedded prime of rank n < m. Then by the Finite Basis Theorem, P/0{P) contains n pairwise disjoint elements of corresponding to elements of P which are not in 0{P), say {&*})t=i C P. Then 6j^ C F for each A; such that 1 < A; < n and G(ULi ^it") ^ ^ S lex(G). (3) =^ (1) : Assume that 7r(G) > m. Then there exists S = {Qj}f=i C Min+(G) which contains m elements and is not ample. Let Q = VS ^ lex(G). For each j, let Qij € Qt \ Qj. Then ft = Vf^^Qij e Qt \ (Uk^iQk)- Disjointify by defining ql = Ai^jQj — AjJLigfc ^ Qi. Then q] €^ Qt for every j ^ i and then we obtain that G'dJ^i Qj^) C V5 g lex(G). Hence, 7r(G) < m. By the minimality of m in (3) and by (1) =>■ (3), we must have 7r(G) =m. (3) «> (4) : Follows directly from the Finite Basis Theorem [D, 46.12] applied to P/0{P) for any embedded prime P. ■ The following is immediate: Corollary 2.5.3. Let G he an i-group. The follovnng are equivalent: 1. 7r(G) < 00. 2. sup{rk(P) : P embedded} < oo. 3. There exists m G N such that lex(G) = Gi\J^^^ af) for all families of m pairwise disjoint elements {aj}^i. We now consider some ^-group-theoretic properties of the prime character. Note that for ^-groups A and B, A^B denotes the ^-group AxB with componentwise operations and is called the cardinal sum. 39 Proposition 2.5.4. Let G he an i-group. 1. For any £-homomorphic image, H, we have i^{H) < 7r(G). 2. For any C E C{G), 7r(C) < 7r(G). //7r(G) < oo, then 7r(C) = 7r(G) if and only if C contains every minimal prime subgroup of G. 3. If'K(A),7r{B) <oo and G = ABB, then 7r(G) < n{A) + n{B) - 1. Proof: (1) Let (p : G -^ H he aji ^-surjection with kernel K. Then by [D, 9.11], the prime subgroups of H correspond to prime subgroups of G containing K. Thus, by the characterization of prime character in terms of ranks of prime subgroups, we have that 7r{H) < 7r(G). (2) This result follows from [BKW, 2.4.7] and [D, 27.8]. (3) Let G = AmB and let P be a prime subgroup of G. Then P = (PnA)ffl(PnB), by [D, 27.8]. Hence P contains at most (m — 1) + (n — 1) = m + n — 2 minimal prime subgroups and therefore 7r(G) < m + n — 1, as desired. ■ Recall from [D, 36.1] that a class C of ^-groups is a radical class if G e C implies the following: 1. ^{G)CC, 2. every ^-isomorphic image of G is in C, and 3. if {Ax}xeA C (tiG) n C, then VacaAa e C. ■-1I;:-; ;-^-' ■ ■■. i^-\.. '■■■ In view of Proposition 2.5.4, it is natural to ask if the class of all ^-groups of finite prime charax;ter is a radical class. The answer is no; the following is a counterexam- pie. Example 2.5.5. We construct an ^-group with two convex ^-subgroups, A and B, each of finite prime character such that the supremum Av B has infinite prime ■^•i. j ■ 40 J- i ? • • i >■ chaxacter. Lb (2.1) Let r be the root system above, where the subgraphs Ta and Tb each have infinitely many identical branches descending from its maximal vertex. Define G = E(r, R), A= {v eG : supp(u) C Ta} and B == {v e G : supp(v) C Tb}. Then we have that A ^ S(r^,R), B ^ E(rfl,R) and ^ V B ^ ^{Fa U Tb,^)- Now, it is evident that 7r(G) = 00, it{A) = 3 = 7r(B), and Tr{A V B) = oo. D Recall that an ^-group G has a finite basis if it contains a finite maximal set of elements {bj}^^i such that the set {g € G'^ : g < bj} is totally ordered for each j. The foUoAving indicates when we can expect 7r{Ay B) < oo: Proposition 2.5.6. Let G be an i-group and let A, B be convex i-subgroups such that iriA) = m < oo and ■k{B) = n < oo. //lex(A V B) = lex(A) V lex(B), then 'k{A\/ B) < 00. Otherwise, it{Av B) < oo if and only if each of A and B has a finite ba^is. Proof: Assume that lex(AvB) = lex(A) Vlex(B) and let P g lex(AVB) be a prime subgroup of G. Then either lex(A) ^ P or lex(P) ^ P, or both. Say \ex{A) ^ P. Then P n lex(>l) is an embedded prime subgroup of A, and since ^{A) = m, we have that rk^(P n lex(A)) < m - 1. Then by [BKW, 2.4.7], rk^vB(P) < m - 1. Likewise, if lex(B) ^ P then rkyivB(P) < n - 1. Thus, for every prime subgroup P g \ex{A V B), we have that rk,i vb(P) < max{m -l,n-l}<oo. Therefore, by Proposition 2.5.2, tt{A V P) < oo. 41 Note that we always have that lex(A) V lex(B) C lex(^ V 5). We assume now that lex(A V B) ^ lex(A) V lex(B) and let P e Spec+(A V B) have the property that lex(A) V lex(B) C P g iex(A V B). If each of >l and B has a finite basis, then rk>ivB(^) = |Min+(^)| + |Min+(J5)| < oo and for any embedded prime subgroup Q ofAVB, we have that iIcavbCQ) < |Min+(A)| + |Min+(B)|. Thus, by Corollary 2.5.3, we have that iriAs/B) < oo. Conversely, 7r(AvB) < oo implies that rkAvB(^) < oo. Hence, since |Min+(A)|, |Min+(B)| < rk^vB(-P)) we have that each of A and B has a finite basis by the Finite Basis Theorem. ■ The proof of the following is evident: Proposition 2.5.7. Let G be an i-group and let A,B E €(G). If A C B = lex(B) or iflex{A) = A and lex(B) = B, then \ex{A V B) = \ex{A) V lex(5). We now compare the property of finite prime character to Conrad's Property F: every element g 6 G"*" exceeds at most a finite number of disjoint elements. The following is compiled in Conrad's Tulane Notes, [C]: Proposition 2.5.8. Let G be an (.-group. The following are equivalent: 1. G has Property F. 2. Every bounded disjoint set in G is finite. 3. For every element g E G, the convex (-subgroup G{g) has a finite basis. 4- Every element of G is contained in all but a finite number of minimal prime subgroups. Corollary 2.5.9. Let T be any root system in which each maximal element lies above a finite number of minimal elements. Then E(r,R) has Property F. Proof: Let v 6 E(r, R). Let C be a maximal chain in F and let the associated minimal prime subgroup be He = {v e E(F,R) : 7 e C =^ z;(7) = 0}. U v ^ He, 42 then there exists 7 e C such that ^(7) 4" 0- Since supp(t;) is finite, v is in all but finitely many minimal prime subgroups of S(r, M). Thus by Proposition 2.5.8, E(r, R) has Property F. ■ Proposition 2.5.10. //G is a finite-valued £- group of finite prime character, then lex(G) has Property F. Proof: Let g G lex(G). Then any minimal prime subgroup of lex(G) not containing g is contained in a value of g. Since each value of g contained in lex(G) contains a finite number of minimal prime subgroups and there are only finitely many values of g, there are only finitely many minimal prime subgroups of lex(G) not containing g. Thus lex(G) has Property F. u , Proposition 2.5.11. Let G be an i-group and let m be a positive integer. If m is minimal such that every pairvnse disjoint subset of G contains at most m — 1 elements, then 7r(G) = m. Proof: Any proper prime subgroup of G is contained in a value of G, hence proper prime subgroups of G contain at most m — 1 minimal prime subgroups. Thus 7r(G) < m. Let n <m and assume that every family of n minimal prime subgroups of G is ample. Then every proper prime subgroup (in particular, every value) of G contains at most n - 1 minimal prime subgroups. This contradicts the minimality of m. Thus 7r(G) = m. ■ Example 2.5.12. The following is an example of an ^-group of finite prime char- acter such that lex(G) has Property F but G has pairwise disjoint sets of any size m. Thus the converse of Proposition 2.5.11 does not hold. Let r be the root system: (2.2) 43 and let G = E(r,R). Then n{G) = 3 and G is finite- valued. Thus, by Proposi- tion 2.5.10, lex(G) has Property F. We demonstrate that there are bounded disjoint families of any given size. Index the maximal elements of T by fj.j, where j is a positive integer, and let Vj e G be such that Vj{fij) — 1 and supp(uj) C {7 6 P : 7 < /ij} where Vj{6) = for all (J < Hj. Then {vj}j>i is an infinite pairwise disjoint family in G. Let a positive integer m be given. Choose any m elements from this set, {vj,, Vj,, . • • , Uj„} and let S = UfcLiSupp(%). Let V be the characteristic function on the finite set S, then « € E(r,R) and u > Ujj for A: = 1,2, . . .m. D Example 2.5.13. The converse of Proposition 2.5.10 does not hold. That is, we present an example of a finite- valued ^-group with Property F and infinite prime character. Consider the following root system F which is indexed by the positive integers: A A (2.3) Then each prime subgroup of the ^-group G = E(r, R) contains a finite number of minimal prime subgroups, yet there is no bound on the number of minimal prime subgroups in each prime subgroup. Thus 7r(G) = 00. G has Property F by Corollary 2.5.9. D 2.6 Filet Character The third character that we define measures the length of a chain of incidents of branching. Definition 2.6.1. Let G be an t-groxip. C = {Pi,Qj G Spec+(G) : i > 0, j > 1} w called a filet chain of prime subgroups i/ Po 2 A 2 ^2 • • • , for all i, Pi || Qi, and 44 Pi+i V Qi+1 C Pi for alli>Q (see below). (2.4) The length 0/ a filet chain is given by 1{C) = max{j G N : 3Qj G C}. // the maximum does not exist, we write 1{C) = 00. The filet character ^{G) is given by <I){G) = sup{/(C) : C is a filet chain}. //Spec+(G) has no filet chains, i.e., if G is semiprojectable, then we say that <I>{G) = 0. Proposition 2.6.2. Let G be an i-group. Then <i>{G) <l if and only if7t{G) < 2. Proof: Suppose that 7r(G) < 2. If (f>{G) > 1 then there exists a filet chain C of length 2 in which we may assume that Pq = lex(G). Since rk(Pi) = 2, we have that 7r(G) > 2, by Proposition 2.5.2. Conversely, assume that 7r(G) > 2. Then there exist minimal prime subgroups P2, Q2 such that Pi = P2 V Q2 / lex(G). Thus for any minimal prime subgroup Qi ^ Pi, the set C = {Pq,Pi,P2,QuQ2}, where Pq = lex(G), is a filet chain of prime subgroups in G of length 2. Therefore <I>(G) > 1. m For larger filet character, the relationship between it and the prime charac- teris more complicated, as the following example illustrates. Example 2.6.3. Let F be the following root system: (2.5) 45 Then G = E(r,R) has <^(G) = 2 while 7r(G) = 4. D The following relationships between the characters hold: Proposition 2.6.4. Let G be an i-group. Then 1. <^(G) < rk(G) - 1. 2. (I>{G) < n{G) - 1 < Tk{G). Proof: Let rk(G) = m. If G has a filet chain C = {Pj, Qj e Spec+(G) : i > 0, j > 1}, then rk{Po) < m and hence 1{C) < m- 1. Therefore, <f>{G) <m-l= rk(G) - 1. If 7r(G) = n and C = {PuQj 6 Spec^(G) : i > 0,j > 1} is a filet chain in G then 1{C) < oo since rk(Pi) < n — 1 by Proposition 2.5.2. Hence, in fact, 1{C) < n - 1 and therefore, 0(G) < n - 1 = n{G) - 1. Now, 7r(G) < rk(G) + 1 by Proposition 2.5.2. Thus, finally, 0(G) < 7r(G) - 1 < rk(G). ■ At this time, any statement that we make about the filet character of an ^-group requires a restriction on the rank and prime character. Rather, we can not say much more than what we establish earlier in this chapter. We leave this investigation for a later date. '■ ^ ■-»-; CHAPTER 3 GENERALIZED SEMIGROUP RINGS Let A be a root system. Starting from the standard semigroup ring and Hahn group constructions and the investigation of the following section, we build an /-ring H having A as the skeleton of the graph of Spec^(7i). We follow up on ideas presented in two papers of Conrad [Cl],[C2], in the paper of Conrad and Dauns [CD], and in the paper of Conrad and McCarthy [CMc]. The first two papers look at the conditions on A which will yield a ring structure on V{A, R) and on the subgroup F{A, R) of elements v for which supp(u) is the join of finitely many inversely well- ordered sets in A. The paper [CD] focuses on the case when V(A, R) is a division ring, while in [CMc] the conditions are established for the ring to be an ^-ring its properties are studied when A is finite. Note that F(A,R) is denoted by W{A,R) in [C1],[C2], and [CD]. 3.1 Specially Multiplicative /-Rings Let r be a partially ordered group which is a root system. Suppose also that r is torsion free, that the subgroup H ofT generated by the positive cone is totally ordered and that T/H is finite. In the paper of Conrad and Dauns [CD, 2.2], it is shown that V{T, R) = F(r, R) and that V{r, R) is a lattice-ordered division ring under the usual group-ring multiplication: for u,v E V(r, R), and 7 € F u * v{'y) = yj u{a)v{l3). It is easy to see that if an element of (V"(r,R),-|-,*) is special (i.e., has only one maximal component), then its multiplicative inverse is also special. Hence, the 46 47 special elements of the ring y(r,R) form a multiplicative group. Moreover, the following holds in general, [CD, Theorem I]: Theorem 3.1.1. Let R be a lattice-ordered division ring with identity. The follow- ing are equivaient: 1. The special elements of R form a multiplicative group or the empty set. 2. Ifa^R^is special, then a~^ > 0. 3. Ifa^R is special, then a~^ is special. 4- For all a e R^ special and x,y e R, a{x V y) = a:cV ay. Since the authors of the paper [CD] seek an embedding theorem for ^-fields, their investigation in this realm is restricted to division rings. In this section, we consider a class of /-rings (which are not division rings) in which the special values form a partially ordered semigroup. The pursuit of a characterization similar to Theorem 3.1.1 of ^-rings satisfying this condition is left for another time. For the /- rings that concern us here, some particular assumptions are needed on the associated semigroups. As a formality, we define: Definition 3.1.2. Call an f-ring A specially multiplicative if the special values of A, with an appropriately adjoined 0, form a partially ordered semigroup. Recall that an /-ring A is called an SV-ringifA/P is a valuation domain (i.e., the set of principal ideals is totally ordered) for every prime ideal P. Let G be the class of commutative /-rings which are local, bounded (that is, A* = A), semiprime, finite- valued, finite rank and square root closed SV-rings with identity and bounded inversion. We demonstrate that the elements of Q are specially multiplicative and then investigate the properties of the associated semigroups. We must first remind the reader of a couple of facts about special values and of the relatively deep theorem, recorded as [HLMW, 2.14]: 48 Theorem 3.1.3. Let A be an f-ring of finite rank with identity and bounded inver- sion which is local, semiprime and square root closed. Then A is an SV-ring if and only if whenever < a <b and b is special, there exists x ^ A such that a = xb. The proof of the following lemma is well-known and routine, however, it is instructive, so we include it here. We remind the reader that we denote the root system of values of an ^-group G by r(G) = {Vj : 7 6 F}, where T is a partially- ordered index set that is order-isomorphic to r(G). Lemma 3.1.4. Let G be an (.-group and let a,b ^ G^ be distinct special elements with values atVa,Vfi, respectively. i- Va II Vff if and only ifaAb = 0. 2. Va < V^ if and only ifa<^b. Proof: First, assume that Va \\ Vp. We show that a A 6 is contained in each value of G and conclude that a and b are disjoint. Let V be a value in G and assume that b ^ V. Then V C V^ and a e V since, else, V C Va, which contradicts the assumption of incomparability. Thus, by convexity we obtain that aAbGV. Again by convexity, if b e V then a A b € V. Thus a Ab = 0, as desired. Conversely, assume that a and b are disjoint and, by way of contradiction, assume (without loss of generality) that V^ < V^. Since a ^Va and K» is a prime subgroup, b ^ Va < V^, a contradiction. Second, assume that Va < V^. Then a e V^. If there exists an integer n such that na > b, then since 6 ^ V/j, we must have that no ^ V^, by convexity. But this contradicts that a e F^, and so we conclude that a <C 6. Conversely, assume that a < 6. We know that b^Va and hence Va < V^. If Va = V^, then by [D, 12.6], there exists an integer n such that na > b, which is nonsense. Thus K* < V^. ■ 49 Combining Theorem 3.1.3 and Lemma 3.1.4 leads to the main result of this section: Theorem 3.1.5. Let A^Q. 1. // a, 6 6 A^ are special and not disjoint, then ah is special. 2. If a, a' are special with value Va, b, b' are special mth value V^ and Va,Vff are comparable, then the special value of ab is the same as that of a'V. Proof: Let a,6 € .A+ be special with values at Va^V^, respectively, and let M be the maximal ideal in A. Since A is bounded, we may assume without loss of generality that Va, V^ < M. This gives rise to the relations a, 6 < 1 and ab <C a, by Lemma 3.1.4(2). Theorem 10.15 of [AF] states that an abelian ^-group is finite- valued if and only if each positive element is a finite disjoint sum of positive special elements. Thus we assume, by way of contradiction, that ab = ci + C2 + he*, where each Cj e A"*" is special and Cj A Cj = for alH / j. Then < Cj < a6 < a for each i and therefore Theorem 3.1.3 gives the existence of Xj e A'^ such that Cj = axi for each i = l,...,k. Without loss of generality, we may assume that < Xj < 6, by replacement with Xj A b. Assume that Xj = YllLi ^ij ^ ^ decomposition of Xj into a sum of special elements. Then Cj = Y^Li o-^i, implies that axj^ = for all but one j, by [AF, 10.15]. Thus we may choose each Xj to be special, without loss of generality. For each i, let V^^ be the value of Xj. Then 6 ^ Vs< and therefore for all i, we have that F^. < V^. If 1^. < V^, then Xj <^ b. This gives a contradiction, since it implies that oxj < ab. On the other hand, if V^. = V^, then there exist n, m G N such that Xj < nh and 6 < mxj. Therefore, axj < nab and ab < maxi. Thus, by [D, 12.6] we have that the value of Cj is the unique value of ab and ab = axj. Hence, ab is special. 50 For the second statement, let Vs be the value of the special element ab. We show that Vs is the value of abf and then a similar argument and transitivity gives that Vs is also the value of a'b', as desired. Towards this end, let V^ be the value of ab'. If V^ II Vs, then = ab Aal/ = a{b A b') and hence = a A (6 A ?/). On the other hand, if Va < V^, then since bAbf ^Va,'we have that a e Va- If V^ < Vq, then since a ^ V^, we have bAb' and hence 6 or ?/ is in VJj. These contradictions lead us to conclude that V* < Vy or V^ < Vs. If Vs < Vy then ab < ab'. However, by [D, 12.6], there exists an integer n such that nb > H and therefore, nab > naii . Likewise, V^ ^ Vs. Thus V^ = V^, as desired. ■ Let i4 6 ^. Then by [AF, 10.10], all the elements of Y{A) are special since A is finite-valued. Abusing notation, we now identify values with their indices and define an operation on P. Let a, 6 e A be special elements with corresponding values at a,/S, respectively. Append <^, a generic symbol, such that (^ || a for all a G F and define a multiplication • on F U {^} such that <f)-a = <!) = a- <f> and /<?i if a I the value of ab oth( \\P, otherwise. By Theorem 3.1.5, this operation is well-defined. Some properties of this operation are recorded in the following proposition. Define a ~ /? if and only if a and /? are contained in the same maximal chains of F. Then ~ is an equivalence relation on F. Let Fq denote the ^-equivalence class of a. Proposition 3.1.6. Let AeQ and let a,j3,'y,5 gT correspond to the values of the special elements a,b,c,d € A"*", respectively. Then 1. The operation is associative, for every a, /?, 7 : (a ■ /?) • 7 = a • (/? • 7) 2. If M is the maximal ideal of A, then M is a value of the identity. Let fi = M. Then fi- a = a- fj. = a for every a < fi. 51 3. Ifa-p^(l)thena'l3<a,l3. 4. Ifa<(3 and 7 is comparable with a, then a • 7 < /? • 7 and 7 • a < 7 • /?. 5. // a / /i, then a • a < a. 6. Maximal chains in F are closed under the operation. 7. For each a € F, the equivalence class Fa is closed under the operation. 8. If'y< a, then there exists /3 G F such that 7 = a • /?. Proof: (1) Note that the products under consideration are all (f) if any of the factors multiplied are actually (^. If a || )9 then aAb = 0. Thus aA&c = and hence a || /?-7 which implies that (a • /?) • 7 = ^ • 7 = ^ = a • (/9 • 7). Likewise, if ;5 • 7 = then both products are equal to <j). Assume that a • /? ^ (^ and /? • 7 / <^. Then either (a • /5) • 7 = or it is the value of the special element {ab)c = a{bc). Thus, it suffices to show that a • /? || 7 if and only if a || ;9 • 7. If a || /5 • 7 then a A 6c = and hence = acAbc= {aAb)c and so = (a A 6) A c. Therefore, for any value 7/ of a A 6, we have tj \\ 7. But a- 13 <r}, so a • /? II 7. The converse follows similarly. (2) Since A is bounded and has bounded inversion, the maximal ideal is a value of the identity, by [Wo, 2.3.4]. Hence, it follows that // • a = a • // is the value of la = al = a, namely a. (3) Assume, without loss of generality, that /3 < a < n. Since ab •< b, we obtain that a •/?</?< a. If /?< a = /i, then a- /3 = fjt- /3 = p <a,hy (2). (4) If a < /9 then a ■< 6 and thus ac '^ be and ca -C cb. By Lemma 3.1.4(2), this just says that a • 7 < /? • 7 and 7 • a < 7 • /3, as desired. (5) Since a^ -C a, we know that a • a < a. (6) Assume, without loss of generality, that /? < a. Let 6 < f3he such that 6 \\ a- p. Then abAd = and < d <C 6. Thus by Theorem 3.1.3, there exists y € A such that 52 d = yb. Then = abAyb= {aA y)b gives that = a A y A 6 and hence = ayb = ad and therefore aArf = 0. This says that a || (J, a contradiction. We now have that a-/? is comparable with any 5 < ^.It follows from this fact and from (3) that maximal chains are closed under the operation. (7) Let 7, (J G Fq. By (6), we have that 7 • (5 is in the same maximal chains as 7 and 6. Hence 7 • 5 e Fq. (8) We know that 7 < a if and only if Vy < Va if and only if < c < a. This implies that there exists a positive element b such that c = ab, by Theorem 3.1.3. We have, by [AF, 10.15], that b is special. Thus, if V^ is the value of b, then 7 = a • ^5, as desired. ■ Corollary 3.1.7. Let A € Q. Then A is specially multiplicative. Moreover, the operation • is a surjection onto T{A) U {<^}. 3.2 r-Svstems and ^-Systems Let A be a given root sj^tem. The discussion of the previous section and of the papers [Cl], [C2], [CD], and [CMc] prepare us to construct an /-ring having A order-isomorphic to its root system of values. However, under the assumptions placed on A in these papers, these rings sometimes are ^-rings, and rarely are /-rings; in fact, as mentioned before, the paper [CD] focuses on the case when y(A,R) is a division ring. Also, contrary to our intentions, [CMc] concentrates on the properties of the ring when the root system is finite. We modify all these conditions on the root system in order to get a ring multiplication, *, yielding an /-ring structure on 71 = (F(A,R),+,*) such that the subring 8? = (E(A,R),-H,*), as defined in Chapter 2, is an /-subring. We start with [Cl, Theorem I] which establishes the ring structure. Proposition 3.2.1. Assume that A is endowed with a surjective partial binary op- eration -I- : A -» A defined on A C A x A. Let u,v €. y(A,R), and define for 53 5 e A, u*v{5) = Y>a+$=s u{a)v{l3). Then both 3? and H are closed under *. In fact, 9? and H are rings if and only if the operation on A is (Baer-Conrad) associative: (a,/3), (a+i3,7) e A if and only if{(3,j), (a,/?+7) ^ A, and if{a,(3), (a+^,7) ^ A then (a + ^) + 7 = a + (/? + 7). Certain properties of these rings, 3? and K, are completely determined by the operation + on A; conmiutativity is one such attribute. Definition 3.2.2. Let (A, +) be a root system with a partial binary operation, +. We call (A, +) an r-system if the operation is surjective and (Baer-Conrad) asso- ciative. An r-system (A, +) is commutative if (a,/?) E A if and only if {f3, a) G A and a + /? = /? + a for all such pairs. Proposition 3.2.3. Let (A, +) be an r-system. (A, +) is commutative if and only iflZisa commutative ring. Proof: If (A, +) is commutative, then for u,v EKwe have u*v{6)= y^ u{a)v{P) = y^ v{(3)u(a) = Y^ v{(3)u{a) = v *u{S). Thus 7i is a commutative ring. ' "' ' ^ '• Conversely, if a + ^5 / /? + a, then Xa+^ = Xa*Xpi' Xp*Xa = X^+a- ■ In order to obtain an ^-ring, we must ask that the operation on A preserve the order of the root system and restrict the domain of the operation. As we will indicate, the various strengths of order preservation and restrictions of domain yield varying richness of order structure on the rings. Definition 3.2.4. Let A be a root system. An r-system (A, +) is an ^-system if it satisfies: a < /3 and (a, 7) 6 A implies that (/?, 7) e A and, in this case, a+7 < /?+7; and »/(7, a) e A implies that (7, f3) E A and, in this case, 7+a < 7+/?. If every connected component of A has a maximal element, we call A bounded above. 54 Note that the ^-system condition gives that no nonmaximal element of such a bounded above root system is idempotent. The following theorem is in [CMc, §2]. Theorem 3.2.5. Let (A, +) be an r-system. Then U is an i-ring if and only if A is an i-system. In the next section, we describe the conditions on an ^-system which induce an /-ring structure on 72.. As mentioned, there are some restrictions that we need to place on the domain and range of the operation. Before we discuss this situation, we present an example of an ^-system that gives rise to an i-xmg which is not an /-ring. Probably, this is the simplest example of such an ^-system. Example 3.2.6. Let A > a > 2q; > Sa . . . and //>/?> 2/? > 3/3 ... . Then we let Aa = {A} U {na}'^^, let A^ = {n} U {n/3}^i and totally-order Aq x A^ lexicographically such that (^i , 71) < {62, 72) if and only if 5\ < 62 or we have 5i = S^ and 7i < 72. Identify pairs with sums and let A^^.^ = {<J+7 : {^, 7) £ A^ x A^}. We define an associative and commutative addition on the root sj^tem given by disjoint union A = Ac U A^ U Aa+/? as shown in the following table, where i^ = A -I- /x and k, I, m, n are positive integers (note that the table is completed by reflection across the diagonal): + a A M V X + ml3 ka + ii na + l0 Q 2a a + /9 a + /i a + /i a + m0 {k + l)a + ti (n + 1)q + 10 /3 2/3 A + ^ fi A + /9 A + (m + 1)0 ka + na + (l + 1)0 A A V V A + m/3 ka + ^ na + l0 (t ^ V A + m/3 ka + fi na + l0 V V A + m/? ka + n na + l0 A + m/3 A + 2m/? ka + m0 na + {m + l)0 ka + ii 2Jta + /i (fc + n)a + 10 na-\-l0 2na + 210 Let u, u e 71'^. We need to show that u*v € 71'^ in order to conclude that Tl is an ^-ring. Since u and v are positive, it is evident that u*v{X),u*v{fj.), u*v{i') > 0. If = u * v{X) = u(A)t;(A), then assume that u(A) = 0. For an integer n, u * v{na) = u(X)v{na) + u{na)v(X) + \^ u{ia)v{ja). i+j=n 55 If u(Tia) = for all n, then « * t;(na) = 0. Conversely, if n ^ is minimal such that na € supp(w), then u * v{na) = u{na)v{X) > 0. Thus, if v{X) > then na is the maximal element of supp(« * v) below A. If 7;(A) = and v{Tna) — for all m, then u*v vanishes on Aq. Otherwise, let m be minimal such that ma 6 supp(7;). Let k = n + Tn then ka is maximal in supp(« * v) and u * v{ka) = u{na)v{ma) > 0. A similar computation demonstrates that w * t; > at the maximal element of its support in A^ also. Assume that = u*v{u) and, when they exist let A„ = mox(supp(u) n Aa) A„ = max (supp(t;) n Aa) /i„ = max(supp(u) H A^) ^ = max(supp(t;) n A^) 5rt = max(supp(u) n Aa+^) (J„ = max(supp(t;) H Aa+0) (3.1) Then the maximal element of supp(M * v) lying below u is given by 7 = max{ Au + //u, Au + 5v, A„ + /iu, A„ + (Ju, Mw + <^t), /^o + <^tt> <5u + Sy} and so we are left to show that u * u(7) > 0. We have three cases to check here, namely 7 = A + Tnj3, ka+fiOTna + ip. Upon consideration of the formulae below, it quickly becomes clear that the only nonzero summands of « ♦ t;(7) are those of the form u(<7)v(r), where a and r are one of the six maximal support elements listed above in (3.1), which implies that the simimands are all positive. For the sake of a certain degree of completeness of exposition, we list the possible summands in the case that 7 = A + m/?. The analysis in the other cases is similar. u*v{X + mp) = u{X)v{ml3) + u{TnP)v{X) + «(A)t;(A + m/?) + u{X + m/3)u(A) + u{n)v{X + mp) + u{X + mP)v{fi) + u{t/)v{X + mp) + u{X + m0)v{y) + Y, u{X + i/3)v{jP)+ Y^ u{i/3)v{X + JP)+ Yi u{X + iP)v{X + jP) i+j=Tn i+j=Tn i+j=m ijjtO «J?tO ijjto To compute u * v{X + mP), consider all the possible combinations of the following situations and then add. 56 1. If «(A) / then if v{mP) / 0, we have that ml3 < /x„. If m/3 < //„, then 7 = A + m/? < A„ + /i„, which is nonsense. Thus m/3 = /i„, in this case and u{Xv)v(nv) is a summand of it * z; (7). 2. By an argument similar to the above, if u(A) ^ and u(m^) 7«^ 0, then mP = /iu and M(/x„)t;(A„) is a summand. 3. If u(A) / 0, w(/i) 7^ or u{v) ^ then if t;(A + mp) y^ 0, we may conclude that 6^ = X + mp and tt(A„)t;((J„),u(/x„)t;((5„) or, respectively, it((J„)u((J„) is a summand. 4. Similarly, if t;(A) / 0, u(/i) ^ 0, or v{i/) ^ then (J« = A + m/3 and u((J„)u(A„), m((Ju)w(/A;), or u{6y,)v{5„) is a summand. 5. If, for some i + j = m, we have that w(A + iP)v{jP) ^ 0, then we also have 6^ = X + iP,fiv= JP and u(5„)u(/i„) is the only nonzero contribution from this large sum. 6. Likewise, iiu{ip)v{X+jP) ^ for some i+j = m, then u{n^)v{6y) is the only nonzero component of this summation. 7. Finally, if u(X+iP)v{X+jP) / then u{6u)v{6y) is the only nonzero summand coming from this summation. As stated before, the analysis in the other two cases is similar. We provide the formulae below: u * v{ka + /i) = u{ka)v{iJ,) + u{iJ,)v{ka) + u{X)v{ka + fi) + u{ka + /i)u(A) + u{ij)v{ka + /x) + u{ka + fji>)v{fi) + u{t/)v{ka + /j) + u{ka + /i)u{i/) + tJ u{ia + iJ,)v{ja) + 22 u{ia)v{ja + n)+ >J u{ia + fx)v{ja + n) i+j=k i+j=k i+j=k iJjtO «j5tO ijjto ■■\ y '1 \ t :i.. , :. - OU; 57 > w * i;(na + Ip) = u{na)v{ll3) + u{l/3)v{na) + u{X)v{na + 10) + u{na + l0)v{X) + u{n)v{na + //?) + «(7iQ! + W)v{fj,) + u{u)v{na + 10) + u{na + I0)v{v) + u{na)v{\ + //?) + u(A + I0)v{na) + u(na + Ai)u(i/?) + w(f/?)u(na + /x) + ^ «(zq; + //?)w(JQ!) + ^ u{ia)v{ja + l(5)-\- ^ u{na + i0)v{j 0) i+3=n i+j=n i+i=i tj#0 ij^O tj?tO + ^ u{i0)v{na + jj5) + ^ M(iQ: + n)v{ja + 10)+ ]^ «(iQ: + I0)v{ja + //) i+j=J i+i=n «+j=Ti JjVO iji^Q ij^ + Y^u{X + ip)v{na + jP) + J^ ^(nQ; + i0)v{X + j/?) + Y^ it(za + A;/3)u(ja + m/?) "R, is not an /-ring: Let ^a„+^ = {u 6 7^ : J € Aq+^ =^ v(5) = 0} be the minimal prime subgroup. Then Xa, X/J ^ ■f^A„+3, yet Xa *X/9 = Xa+/3 ^ -f^A„+^. Thus, ^Aa+/3 is not an ideal and 71 is not an /-ring by [BKW, 9.1.2]. D The following two propositions record some consequences of the ^-system condition. Note that none of the excluded conditions occur in Example 3.2.6. Proposition 3.2.7. Let (A, +) he a bounded above (.-system in which maximal ele- ments act as an additive identity on elements below it. Then the following can not occur for nonmaximal elements a, /? G A, 1. P -\- a < a, where 0, a are below the same maximal element, and \\ -{- a 2. 0<a<0-\-a 3. p<0-l-a<a 4- If X, n € A are maximal and (A, /x) e A, then X -\- fj, it X, fx. 58 In particular, the second and third properties imply that if a, 13, p+a are comparable, then fi + a <a,p. Proof: U 13 + a < a, 13 \\ P + a, and ii> a, pis maximal, then xp, X/* - Xa are positive. Yet, x^ * iXn - Xa) = Xff - Xff+a is negative, contradicting that 7e is an £-ring. Assume that /? < a < /? + a. Let /x > ;5 + a be maximal. Then as above, X/3 * iXn ~ Xa) < 0- The same contradiction is obtained in the case that we assume P<l3 + a<a<n. If A,/iaremaximaland A + /i </i thenxA*(X/t-2xA+;i) = -Xa+m < 0- ■ Proposition 3.2.8. Let (A, +) be an i-system. 1. There is no nonmaximal 5 G A such that 5 is idempotent and 6 + a = 6 for all nonmaximal a> 5. 2. If 6 E A is nonmaximal and idempotent then there does not exist a maximal element ijl> 6. Proof: If (5 e A such that 6 is idempotent and 5 + a = S for all nonmaximal a> 5, Then we contradict the assumption that 71 is an ^-ring since xs * (Xq ~ ^Xs) — —Xs- Ji fi> 6 is maximal and 6 is idempotent, then xs * (X/i — ^Xs) = —Xs- ■ 3.3 f-Svstems We are much more interested in the /-ring situation. In [CMc, 2.1], the authors demonstrate a condition on A which will give rise to an /-ring. Theorem 3.3.1. Let (A, +) be an (.-system. Then It is an f-ring if and only if the root system also satisfies: i/a || /? and {a,^) e A, then a + 'y \\ 13; and 1/(7, a) G A, then 7 -I- a II y3. 59 In this section we consider a condition on A which is equivalent to the one stated above and proceed to investigate certain properties of the associated /-rings. Recall that r(F(A,R)) is order-isomorphic to A by Proposition 2.1.1, where the values are of the following form, for 6 E A: Vs = {uen:'y>S=^ «(7) = 0}. Definition 3.3.2. Let (A, +) be an i-system such that 5 = a + /3 implies a,j3>5. Then we say that (A, +) is an /"-system. Proposition 3.3.3. // (A, +) is an f^ -system, then % is an i-ring and for every (J G A, the subgroup Vg is an ideal. Hence, in particular, 71 is an f-ring and 3? is an f-subring of H. Proof: Assume that (A, +) is an /"-system. Let 6 € A,v eVs^ueH, and j >S. Let 7 = a+(3, then by the /"-system assumption, ^ >'y>6 and hence v{P) = 0, for aJl such P since w G V^. Then u*v{'y) = X)a+/9=7 u{a)v{P) = and we conclude that u*v eVs- Therefore, the values oCR. are ideals. Moreover, since eadi minimal prime subgroup is an intersection of a chain of values by [D,10.8], each is an intersection of a chain of ideals; thus each minimal prime subgroup itself is an ideal. Finally, we have shown that 7e and 3? are /-rings by [BKW, 9.1.2]. ■ Proposition 3.3.4. An t-system (A, -I-) is an f^ -system if and only if K is an f-ring. Proof: Sufficiency is shown in Proposition 3.3.3. Conversely, assume that there exist a,/3 E A such that a-f-/?^aorQ;-|-^^/3. Then by Proposition 3.2.7, we have that a, /3, a + /9 are not all comparable. If a \\ /?, then we have that Q\\a-\- (3 by Theorem 3.3.1. Assume that 13 \\ a-\- P and let C be a maximal chain in A containing a-\- p-Lei Hc = {v €%: 5 EC ^ v{8) = {}i}\ie the associated minimal 60 prime subgroup. Then X0 ^ He, yet Xq * X/9 = Xa+ff i He- Thus He is not an ideal and hence Tl is not an /-ring by [BKW,9.1.2]. ■ Definition 3.3.5. An f^-system satisfying the following is called an /-system; l.Ifa and {3 are comparable, then {a,P), {P,q) G A. 2. If IX is maximal, then {S, /x), (/x, 6), {n, fi) € A and 5 + n = ii + 6 = Sfor every 6 < fi. In particular, /x -I- // = ju. We will short;ly see that these additional assumptions on a bounded above /°- system make maximal chains in A into monoids. This is quite useful in our setting. For instance, it is not difficult to figure out when the /-rings have a multiplicative identity, under the /-system assumption. Proposition 3.3.6. Let (A, -I-) be a bounded above f -system. 8? and % each have a two-sided multiplicative identity if and only if A has a finite number of connected components. Proof: Let (A, +) contain only a finite number of connected components with maximal elements {//i, /i2, •■ • , Mn}- Let Xj be the characteristic function on the set {fij} and let e = Y,%i Xj- Then for w 6 71 and 5 G A, V * e{6) = ^ v{a)e{f3) = v{6)e{fjtk) = v{6), where /Lt* > (J is maximal. Likewise, e*v{6)= Y^ e{a)v{0) = e{iJik)v{6) = v{6). Thus e is a two-sided multiplicative identity. Conversely, assume that e G 7^ is an identity and let /x G A be maximal with characteristic function Xn- Then 1 = X/iCa*) = e* Xm(/^) = e(/i). Thus, supp(e) has * ST 61 as many maximal elements as there are maximal elements in A. Since e G 72., we must conclude that there are only finitely many connected components in A. ■ It is handy to have the following definition: Definition 3.3.7. Call an r-system (A, +) unital if the ring % has a multiplicative identity. .^ - • ^ The rings 3? and "R, are semiprime, as we will now show. Thus the minimal prime subgroups are also prime ideals, by [BKW, 9.3.1]. Proposition 3.3.8. Let (A, +) be an f^ -system. Then K is semiprime. Proof: Let u e 11, where u = Y^j^iCtjXSjj for an index set / of supp(«) and for ttj e R\0 for all j 6 /. Then u*u{6) = Y^s=s.^s ^»%"- ^^^ '^« ^® maximal in supp(u). We show that Si + Si is maximal in supp(u * u). Let S = Sj + Sk ^ supp(u * u). If Sj II Si, then by the /"-system condition, 5 || (Jj + Si. On the other hand, if Sj, Sk < Si, then S < Si + Si,as desired. Thus u * u{Si + Si) = af > and we conclude that Tl is semiprime. ■ Corollary 3.3.9. Let (A, +) be an f^ -system. Then maximal chains in A are closed under the operation +. Proof: Let C C A be a maximal chain and let He — {v € H : S e C =^ v(S) = 0} be the associated minimal prime subgroup. Let a,/? e C and assume by way of contradiction that a + /? ^ C. Then Xa*Xfi — Xa+fi € He but Xa,X$ ^ He- Thus He is not a prime ideal. This contradicts Proposition 3.3.8, by [BKW, 9.3.1]. ■ One should ask if the maximal ^-ideals of TZ are actually the maximal ideals; or equivalently, one should ask if "R has the bounded inversion property. The answer is yes, if (A, +) is a unital /-system. 62 Proposition 3.3.10. Let (A, +) 6e a bounded above unitd f -system. An element ueU is a multiplicative unit if and only ifu^Vf^ for all maximal /i e A. Thus 71 satisfies the bounded inversion property. Proof: Let 5" C A be the set of maximal elements of A and let e = Zl^es Xit be the multiphcative identity. If u e 72. is a multiplicative unit, then there exists v eK such that u*v = e. Hence 1 = e()u) = u{fjL)v{fj.) for every /x 6 5". Thus «(//) / for all // G 5 and hence u ^ V^ for all maximal /x e A, as desired. Assume that u ^ V^ for all maximal /i e A. We define the multiplicative inverse t; of u recursively on each maximal chain in the support of u. Let v{5) = for all 6 ^ supp(m) and let t;(/x) = 1/m(/x) for each maximal /x € A. Then ifSi<n is maximal in supp(n), we just solve the equation = u*v{6i) = u{Si)v{iJ,) + u{fi)v{6i) to get that v{Si) = -u((Ji)/u(/x)^. Proceed with the definition of u accordingly. That is, let fi> S e supp(u) and assume that i;(7) is defined for all 7 > 5. Then u{a)v{T) is a summand of u * v{6) only if (5 < a, r e supp(u). Thus v{5) is the only unknown in and is the unique solution of the equation = u*v{6). Hence, since if e is the multiplicative identity of 7Z and u > e then u ^ V^ for all maximal /x, we have that u is a multiplicative unit. Therefore K satisfies the bounded inversion property. ■ Let (L, <) be a partially ordered set. Recall that A C L is called closed if {ajig/ and AjCj or VjCi exists in L then AjOj, Vjaj e A. It is the case that, if (A, +) is an /-system, then all of the closed convex ^-subgroups of 7^ are ring ideals. We will use the following special case of [D, 45.26]. Theorem 3.3.11. Let G be a finite-valued £- group. Then there is an order-preserving correspondence between the closed convex i-subgroups of G and (order) ideals $ of the root system T{G) given by K t-^^K = {Gs e r{G) •.3k e K such that Gs is a value of k} 63 ^ i-^ K^ = {g € G : all values of g are in $}. Proposition 3.3.12. Let (A, +) be a bounded above f -system. Any closed convex i-subgroup of TZ is an ideal of 7Z. Proof: Let A' be a closed convex ^-subgroup of H; let it G /('+ and t; G 72.. If u is a unit, then $k = r(G) and by the preceding theorem, K = G. So assume that u is a nonunit. Let /x G A be maximal and assume that u * v{'y) = for all /x > 7 > 5 and u*v{5) ^ for some 6. By the preceding theorem, we need to show that S G $«- C A. Let 6 = a-\- 13 such that u{a)v{P) ^ 0. Then there is a maximal a' > a such that u(a') ^ 0. Then a' = a' + fj, > a' + p > a + p = 5 md a' E ^k since u e K. Therefore, we conclude that 6 G ^k since ^k is an order ideal. Hence, u*veK by the preceding theorem. ■ We now seek the prime ^-ideals and 2-ideals among the prime subgroups Vg and their associated value covers. The cover of Vs is the set Ps = {ven: v{j) = for all j > 5} and is the smallest convex ^-subgroup properly containing Vs and xs- First let us recall the most general definition of 2;-ideal. Let G be a vector lattice, v G G"*", and let {^n}^i C G be a sequence. Recall that the sequence converges relatively uniformly to g €. G along the regulator v and write gn — > g, if for every e > there exists iVe > such that for all n > iV^, we have that [^ — ^nl < ^v- Let H he a, convex ^-subgroup (sub-vector lattice) of G. The pseudo-closure of H is H' = {g € G : ^gn}n=i ^ H,gn -^ g for some v G H+}. Then H is relatively uniformly closed ii H — H'; let H denote the relative uniform closure of H. Then if G{g) denotes the convex ^-subgroup (sub-vector lattice) of G generated by g, we define a convex ^-subgroup (sub-vector lattice) if to be an abstract z-ideal if 64 he H,g eG md G{g) = G{h) imply that g e H.ln fa^t, [HudPI, 3.4] says that this definition is equivalent to G{h) C H for all he H. Proposition 3.3.13. Let (A,+) be an f -system. If fi G A is maximal, then V^ is a prime ideal of H which is an abstract z -ideal. Proof: Let u * v €. V),, and u ^ V^. Assume without loss of generality that w+jV"^ € Vft and u~ ^ V^. We show that v~ £ V^. Since u~ ^ V^, we have that «"(/i) ^ 0. Thus since u"*", V^ eVft, a+fi=tt a+0=ft and since /i is maximal, we have that = u~(/i)u~(/i) and therefore, v~{n) = 0, This gives that t;~ e V^ and u € V^. Hence V^ is a prime ideal. Since Tl/V^ is isomorphic to R via the evaluation map u i-> u(/i), we have that V^ is uniformly closed by [HudPI, 2.1]. Thus by [HudPI, 3.4], V^ is an abstract 2;-ideal. ■ Corollary 3.3.14. 7f (A, +) is a bounded above unital f -system, then the maximal ideals are given by {V^ : // 6 A is maximal}. Proposition 3.3.15. Let (A, +) be an f -system and let 6 e A be nonmaximal Define Ps = {v €1I: v{'y) = for all 7 > 5}. Then Ps is an abstract z-ideal and it is a prime ideal if and only ifa-^j3>6 for dla,^ > S. Proof: Let v € Pg. Then there exists {u„}^i C Ps such that t;„ -^ v, for some w G TZ."*". Let 7 > (J, then for every e > we have that |u|(7) < 610(7). Thus ^(7) = and Ps is relatively uniformly closed. Therefore, Ps is a 2-ideal by [HudPI, 3.4]. Assume that Ps is a prime ideal and that there exist a,0 > 6 such that a + 13 < 6. Then Xa+0 G Ps and Xa+0 = Xa * Xp- But Xa.Xp i Ps, which is a contradiction. Conversely, assume that a+ P > 5 for all nonmaximal a,/3 > S. Let u*v £ Ps and assume, by way of contradiction, that u,v ^ Ps- Then u * v{^) = for all 7 > (J and there exist elements a e supp(u) and ^ e supp(t;), each maximal in the support set and such that a,P > S. Assume, without loss of generality that a > /?. If o^ <a,iy < and at least one of the inequalities is strict, then a' + /?' < a + ^ and we conclude that u*v(a + l3) = u{a)v{l3) + u{f3)v{a). If a = /?, then since a + /3 > 5, we have that = u*v{a + /3) = 2u(a)v{/3) ^ 0, a contradiction. If a > /3, then v{a) = and hence = u* v{a + P) = u(a)v{P) / 0, another contradiction. Thus we conclude that either u or t; is in Ps and therefore Ps is a prime ideal. ■ Let 5 be a totally-ordered set. Recall that S is an rji-set if whenever A,B C S are countable and A < B, then there exists c G 5 such that A < c < B. Since R is not an 771 -set, the ring H is never an 771 -set. To see this, let 5 G A and consider the sets {xs} > {(1 - ^)xs '■ n G N}. But, H can satisfy a related, slightly weaker condition. Definition 3.3.16. We call a totally- ordered set S an almost rji-set if A,B C S are countable and if A < B, then there exists cE S such that A<c<B. Note that E is such a set. Proposition 3.3.17. Let (A, -h) be a totally- ordered f^ -system. H is an almost Tji-set if and only if A is an rji-set. Proof: Assume that 71 is an almost 771-set. We first note that A contains no suc- cessor pair. Let a < /? be a successor pair. Then the sets {nxa}n&i and {l/nX/3}n6N contradict the almost r/i-set condition. Let A = {Q!j}jeN,-B = {Pj}jeJi C A, where ai < a2 < ■■■ < Ih < Pi- Then Xai < Xa2 < • • • < X/?2 < Xffi and there exists f ell such that x<m < f < XPj for all i,j. Let 7 be the maximal element of supp(/). Then f — Xai > implies that 7 = ttj and /(7) > 1 or 7 > a^ and 7(7) > 0. If 7 = a^, then / - Xa+i < 0, which 66 is a contradiction. Thus 7 > a^ for all i. Similarly, 7 < /?, for all j. Therefore, A is an r^-set. Conversely, assume that A is an r7i-set and let /i < /2 < • • • < 52 < 5i ^ ^• Let ^1 < 02 < * * • < 72 < 7i ^ A be the corresponding maximal elements of the support sets. Let $ = {(j>j}jefi a^^d T = {7j}ieN- We have a few cases to consider: $ and r are the same constant sequence, one of the sequences is eventually constant, or neither sequence is eventually constant. If there exists n € N such that (ffj = a = ji for i,j > n, then we get the following string of inequalities in R : /i(0i) < f2{<h) < '• < 52(72) < 5i(7i)- Since R is an almost r/i-set, there is r G R such that fj{^j) <r < Qiiji) for all i,j. Then fj < fXa < 9% for all i, j. If $ is eventually constant and F is not, say (f>j = a for all i > n, for some n, then, by hypothesis, there exists /? e A such that a < /? < F. Then /_, <Xp^9i for a,ll i, j. If neither sequence is eventually constant, then by the t/i-set hypothesis, there exists ^ such that 0j < ^ < 7^ for all i and fj < Xff < 9i for all«,j. ■ 3.4 Survaluation Ring and n*^-Root Closed Conditions Recall that a commutative ring ^4 is a survaluation ring (or SV-ring) if A/P is a valuation ring for every prime ideal P. In this section, we set down a character- ization of those /-systems which give SV-rings. Let (A, <,-(-, /i) be a totally ordered and cancellative abelian monoid with identity element fx. We define the group of differences, qA, of A as it is done in [Fu, X.4]. Define an equivalence relation on A x A by (61,52) ~ (71,72) if and only if Si + j2 = ji + S2. It is clear that the relation is reflexive and symmetric; the transitivity follows from the cancellation property of the monoid. We let qA be the quotient A x A/ ~, denote the class of the element (61,62) by [61 - 62], and define an operation + as is usual. That is, [61 - 62] + [71 - 72] = [(61 + 71) - ((J2 + 72)]- The 67 cancellation in the monoid ensures that the operation is well-defined. The element [^ - /i] is an identity and [^2 — (Ji] is an additive inverse of [5i — S2]. We define [Si - 82] < [71 - 72] if and only if (Ji + 72 < 7i + 82. By [Fu, X.4.4], this is the unique order on gA which extends the order on A. Finally, o-embed A in 9 A via Let ff be a partially ordered groupoid. Then h € H is called negative if hx < X or xh < x, or both for all x e H. The groupoid H is called negatively ordered if every element is negative. Definition 3.4.1. Let A be a partially ordered semigroup. A is called inversely naturally ordered 1/ it is negatively ordered and 6 < a implies that there exists /? € A such that (J = a + /?. Example 3.4.2. Let A = {1 - ^}^i U [l,oo) C R be inversely ordered with the usual addition in the reals. Then A is an /-system which is not inversely naturally ordered. To see this, note that (1 — ^) + c = 1 if and only if c = |,ti = 2 or c=l,n=l. D Let (A, -f-, //) be a totally-ordered abelian cancellative monoid with maximal element /i, such that (A, -I-) is an /-system. Let X = {x* : <J € A}. Then (A", *,Xii) is a totally-ordered abelian monoid which is ^-isomorphic to (A, -|-,/i). Since X is written multiplicatively, the elements of the group qX are quotients and we denote them as such in the following proof. Theorem 3.4.3. Let (A, +, fx) be a totally- ordered abelian cancellative monoid with maximal element fx. Let X = {xs : 6 e A} and TZ = F(A,R). The following are equivalent: 1. TZ is a valuation ring. r; '-; --■ . ■ . "i r 1-^*'^":-,.) 2. % is 1-convex. 68 S. His Bezout. 4. V, is convex in qH. 5. X is convex in qX. 6. A is convex in qA. 7. A is inversely naturally ordered. Proof: The equivalence of (1), (2), and (3) is [MW, Theorem 1]; the equivalence of (2) and (4) is [ChDi2, Lemma 2]. (2) =i> (5) : Let X7 < xjxp < Xs- Then < %„ < Xs+fi and by (2), there exists fen such that Xa = f * Xs+p- ^ f = JLjejfi^<t>p ^^^^^ "^ ^ ^^^ ^°^®^ ^®*' fj e R and 0j G A for all j e J, then ifj if' [0 els Thus, for some j G J, we have that a = /? + <J + (^j, /j = 1 and fk = Q for all A; 7^ j. Hence / = x^^ and Xa/X/3 = X(J+*, £ A", as desired. (5) <=> (6) : Follows since A^X. (5) =^ (7) : Let a < ^ e A. Then a + ^ < a and therefore Xa+/3 < Xa < Xi9 which implies that Xa < Xa/Xfi < X/x- Thus, by (5), we have that Xa/Xfi = X6 ^ X for some 6 e A. Therefore, Xa = X^+s and a = f3 + S. (7) => (2) : Let < u < v £ TZ, and assume that u = YljeJ^J^"'} ^^'^ *^^* V = Y!,k£K ^kX0k fo^ index sets J, iif, and where Oj, 6^ G R for all j G J, A; G iiT. Also assume that ai < Pi are maximal elements in the respective support sets. Then by (7), for every j G J and every k e K, there exists 6j,jk G A such that Qj = Pi+ Sj and pk = pi+ 7fc. Note that 71 = //. Then u = X/9i * Z)jej OjX*, and 69 Let w = Ylk^K^kXik- ^^^^ w{n) = bi > and hence ^w is invertible in U. Let X = Y^j^j ttjXSj ■ Then by cancellation in A, we have that < x < u; and < ^x < ^w. If we let / = (^w)~^ * (^x), then ^x = j^w * f ajad x = w * f. Finally, it = X/3i * x = X;8i * i« * / = u * /, as desired. ■ The following lemma is well-known and routine to verify. Lemma 3.4.4. Let (A, +) be an (.-system andC C A, a maximal chain. Denote the associated minimal prime subgroup by He = {u ^ Tl : C Ci supp(«) = 0}. The map (p : F(A, R) -^ F{C, R) given by restriction is a surjective i-ring homomorphism mth kernel He. Thus, n/He = F{C, R) Corollary 3.4.5. Let (A, +) be a unital f -system. Then Ti is an SV-ring if and only if each maximal chain in A is inversely naturally ordered. Example 3.4.6. 1. If Ai = [0, oo) C R is inversely ordered with the usual addition of real numbers, then F(Ai,R) is an SV-ring. 2. If A2 = {1 — -}^i U [1,00) C R is inversely ordered with the usual addition in the reals, then F(A2,R) is not an SV-ring. 3. Let A = R[[x, y]] be the ring of formal power series in the indeterminates x, y. Order the monomials lexicographically via 1 » x, y and xV < x*y' if and only if A; < z or A; = i and I < j. The ring A is not an SV-ring since it is not 1-convex: note that < y < x and the equation y = xf has no solution f eA. - - V . . , ... ' .' Let Zx = Zj, = {n e Z : n > 0} be inversely ordered. In the lex-order described above, if A = Z^ x Zy, then A = F(A, R). We convexify A in qA by convexifying A in qA. That is, if A'' = A U {(n,m) e Z x Z : n > 0,Tn < 0} then ^(A^ R) ^ A{{xY : i > or i = and j > 0}) U A is an SV-ring which is the convexification of A in qA. D 70 Recall that an /-ring A is n^-root closed if for every a ^ A^ there exists h ^ A such that a = 6". This property arises in K if there is a certain amount of divisibility in the arithmetic structure of A. Definition 3.4.7. Let (A, +) he an r -system. A is coiled 7i-divisible if for every 6 e A there exists a G A such that na — 5. We say that the system is divisible if it is n-divisible for all n EN. Proposition 3.4.8. Let (A, +) be a totally ordered f -system such that 71 is n^^-root closed. Then A is n-divisible. Proof: Let 5 G A. Then xs = w" for some v = Ylj^jO'jXaj- li S = fj. is maximal in A, then 5 = nfx, so we assume that (J / /x. If ai € supp(t;) is maximal in the support set, then nai e supp(t;") is maximal. Therefore 6 = nai, as desired. ■ Proposition 3.4.9. Let (A, +) be a totally ordered inversely naturally ordered f- system with maximal element fi. If A is n-divisible, then 72. is n^-root closed. Proof: We begin with square-roots. Let u G TV^ be given by u = X^.^j 0,3X0, for some index set J, Uj G R, and «_, 6 A for all j G J. K u{ii) ^ 0, then we define a square-root v recursively on A. To begin, let v{n) = yju{n) and assume that ai is maximal in supp(u) \ {/x}. Let v{S) — Q for all ai < (J < /i. Then we define v{ai) = u{ai)/2v{n). Let S be the N-linear span of supp(u) and define u((5) = for alH ^ 5. If ^(7) is defined for all 7 > (J, then we define v{5) to be the unique solution of the equation u{5) = ]C(r+T=<5^(^)^ (''")' where, necessarily, a,T E {a e S : a> 6}. Then u — v*v,hy construction. Now, assume that u(/i) = and let ai be the maximal element in the support of u. Since A is inversely naturally ordered, for every j 6 J, there exists Sj e A such that aj = tti -^Sj. Note that 61 = //. Then u = Xai *T>jeJ ^jXSj • Let w = "^.^j ajxsj , then w{ij,) ^ 0. Thus w = vi*vihy the above construction. Since A is 2-divisible, 71 ai = 271 for some 71 £ A. Therefore Xa = X-n * X-n ■ Letting u = X71 * ^i' ^e then have that v*v = Xyi*Vi*X'n*'Vi= Xai*w = u. Thus n*''-roots exist when n is a power of 2. Let n be odd and let w € 71 be given by t; = X)fc€Ar ^^X^k ^^^ some index set K, bk € R, and /?*; € A for 8ll k e K. As with square roots we consider two cases. First assume that v{fi) =^0 and define an n^'-root recursively. Let w{fj,) be a real n*''-root of u(/x) and let v{6) =0 for all ^i < 6 < fi, where pi is the maximal element of supp(t;) \ {/i}. Then the Tvfold convolution product equation we must solve reduces to v{Pi) = w"(/?i) = n{w{n))^~^w{f3i). In order to see this, we proceed by induction on n. If n = 2, then w*w{Pi) = w{tj)w{Pi)+w{Pi)w{fi) = 2w{fj,)w{l3i). Assume that the statement holds for n = m. Then t/;'"+^(A) = {w * w"'){/3i) = w{i^)w"'iPi) + t/;(A)ti;"'(/i) = wifi)miw{fM)r-'w{l3,) + «;(A)(w(/i))'" = (m + l)(«;(//))"*w;(/3i) as desired. We may then define w(/?i) = ^(/^i)/("(^(m))''~^)- In general, we let ^(7) = if 7 is not in the N-linear span of the support of v. Assume that 10(7) is defined for all 7 > (J. We show that the equation v{S) = w^{5) is linear in 'w{5); hence, we may define w{6) to be the unique solution of this equation. If n = 2, then w'^{S) = Y!,a+T^s w{a)w{T) + 2w{5)w{fj,). Assume that w^{6) is linear in w{6). Then «,"*+! (,J) = ^ w{(t)w"'{t) + w{jji)w"'i6) + w{5)w"'{fi) ff+T=S j is linear in w{6), by induction, since w{5) will not appear in w"*{t), as S <t and A is an /-system. Thus, in this case, v has an n*''-root. Second, assume that v{fi) =0 and proceed as in the square-root case. Let /?i be the maximal element in the support of v. Since A is inversely naturally ordered, for every k £ K, there exists 5k £ A such that fik = Pi + Sk- Note that 61 = //. Then 72 ^ = X/J, * 'ZkeKhXs,- Let x = Y^keK^Xs,, then x{n) ^ 0. Thus x = w^, for some wi, by the above construction. Since A is n-divisible, /?i = n7i for some 71 e A. Therefore X/9, = (Xti)"- Let to = X7, * Wi; then lu" = (X7, * Wi)" = X)3, * a; = t^. ■ Corollary 3.4.10. Let (A, +) be a totally ordered inversely naturally ordered f- system with maximal element fx. A is divisible if and only ifKis n^^-root closed for alln. Example 3.4.11. 1. Let Ai = [0, 00) C R be inversely ordered with the usual addition of the real numbers. Then F(Ai,R) is n*''-root closed for all n. 2. Let A2 = {n € Z : n > 0} be inversely ordered. Then A2 is not 2-divisible and Xi > has no square-root. That is, if Xi = v^, then v{0) = and we then conclude that 1 = Xi(l) = 2v{0)v{l) = 0, a contradiction. 3. Let A3 = {1 - -}^3 U [1, 00) C R be inversely ordered with the usual addition in the reals, then A3 is not inversely naturally ordered and similarly, xi has no square-root since 1 has no nonzero summand. D Proposition 3.4.12. Let (A, -I-) be a totally ordered f -system with maximal ele- ment fi. Then if n has an immediate predecessor tt, then Xv has no square-root. If there exists /i > (J G A such that 6 has no nonmaximal summand other than itself, then xs has no square-root. Based on the preceding examples and results on n*''-roots, we formulate the following: Conjecture 3.4.13. Let (A, -I-) be a totally ordered f -system with maximal element /i. F(A,R) is square-root closed if and only if A is 2-divisible and every nonmaximal element of A has a nonmaximal summand other than itself. 73 Recall that a field K is real-closed if every positive element is a square and every polynomial p € K[x] of odd degree has a root in K. An integral domain R is called real-closed if qR is a real-closed field. Let (A,-|-) be a totally ordered inversely naturally ordered /-system with maximal element fj,. Assume that A is also 2-divisible. Then 7^ is a 1-convex and square-root closed /-domain. By [ChDil, Theorem 1], under these conditions, 72. is real-closed if and only if every odd degree polynomial over H has a root. What additional assumptions on A are necessary to guarantee the real-closed property? Conjecture 3.4.14. Let (A,-|-) be a totally ordered commutative f -system with maximal element //. // A is divisible and inversely naturally ordered, then TZ is real- closed. We end this section by shedding a little light on this conjecture. Recall from [HLM] that a commutative /-ring A with 1 satisfies the Intermediate Value Theorem for polynomials (or is an IVT-ring, for short), if for every p{t) 6 A[t], and pair of distinct elements u,v E A such that p{u) > and p{v) < 0, there exists w € A such that p{w) = and uAv<w<uWv. We show that a totally ordered commutative semiprime valuation /-domain with identity is real-closed if and only if it is an IVT- ring. It is necessary to record the following unpublished theorem of Suzanne Larson, which was communicated via electronic mail on April 17, 1997. Her proof follows. Theorem 3.4.15. Let A be a commutative semiprime IVT-ring tvith identity. If S is a multiplicatively closed subset of regular elements of A'^, then the ring of quotients, S~^A is an IVT-ring. Proof: Let p{t) e S-'^A[t] be given by p{t) = oo6o ^ + aiK^t H h a„6-4" and assume that p(itii;f ^) > and p(u2U^^) < 0. Let d = v^v^bobi • • •&„. Then d e 5 is regular. Define a new polynomial q{t) e A[t] by oov^v^bi • . • bn+aiv^-^v^-%b2 • • • 6„<+02<"M~%&i^ • ■ • bnt'^+- • •+anbobi • ■ • 6„_ir. 74 Then q{uiV2) = dp(uit;f ^) > and q{u2Vi) = dp(u2U^^) < 0. Since A is an IVT-ring, there exists w E A such that uiV2 A U2V1 < w < U1V2 V U2V1 and q{w) = 0. Then UiVi^ Au2V2^ < wVi^V2^ < uiVi^ Vu2^^j\ and dp{wv^^V2^) = q{w) = 0. Since d is regular, p{wVy^V2^) = and we conclude that the quotient ring is an IVT-ring. ■ Proposition 3.4.16. Let A be a totally ordered commutative semiprime valuation f -domain with identity. Then A is real-closed if and only if it is an IVT-ring. Proof: If A is real-closed, then qA is a real-closed field and, by [ChDi2], qA is an IVT-field. Let p{t) G A[t] be such that p{u) > and p{v) < 0, for some u,v E A. Then there exists w e qA such that p{w) = and u Av < w < uV v. Since A is a valuation domain, A is convex in qA by [ChDi2, Lemma 2]. Hence, w E A and we have that A is an IVT-ring. Conversely, if >1 is an IVT-ring, then qA is an rVT-field, by the preceding theorem of Larson. Then, by [ChDi2], qA is real-closed and therefore A is real-closed. ■ t.-' ' .' \ -♦ CHAPTER 4 RAMIFIED PRIME IDEALS In this chapter we expand on the notion of a ramified prime ideal, as defined in [Le], which we introduced in Chapter 2. We first examine the concept in general and then move to try to understand ramified maximal ideals which correspond to nonisolated G^-points. This result is then used to consider local versus global ramification conditions. 4.1 Ramified Points Definition 4.1.1. Let X be a completely regular space. A prime ideal ofC{X) is ramified if it is the sum of the minimal prime ideals that it contains. We define p £ X to be ramified if Mp is ramified. A point p G X is totally ramified if every prime z-ideal contained in Mp is ramified. The space X is (totally) ramified if every nonisolated point of X is (totally) ramified. A ramified ^-ideal of C{X) is a prime ideal, by Corollary 2.2.6. LeDonne proves that a ramified prime ideal of C{X) is necessarily a 2;-ideal. Let us consider two extreme conditions. Recall that we say a point p G X is an F-point if Op is prime. If p is an F-point, then since Op is the unique minimal prime ideal contained in Mp, Mp is not ramified. Likewise, in this case, no prime 2-ideal contained in Mp is ramified. On the other hand, the condition of total ramification ensures branching at every prime z-ideal. Aneilytically, LeDonne shows [Le,§ 3]: Theorem 4.1.2. IfX is a metric space then every maximal ideal ofC{X) is totally ramified. 75 76 Note that this result says that every maximal ideal of C{X) (fixed or free) is ramified, if A" is metric. We do not know of any weaker topological condition which guarantees total ramification of C{X). Definition 4.1.3. Let A be a commutative f-ring with identity. For any integer n>2, call a prime £-ideal P n-limbed if P is the sum of n noncomparable prime i-ideals of A which are properly contained in P. A point p of X is n-limbed if Mp is n-limbed. Note that any n-limbed i-ideal P is necessarily ramified and rk(P) > n. Example 4.1.4. We now present an example of an /-ring 11 in which a maximal ideal is ramified but not Tvlimbed for any n. Let A© = [0, oo) C R and define Ajj = (l/n,oo) C R, for i = 1,2, where each interval is inversely ordered. Let A„ = A^ U A^ be the disjoint union and then let A = A^o U (n„ea,An). We obtain a root system with the induced ordering which we describe in the following diagram: (4.1) We endow A with a partial commutative associative binary operation. Let (*)^ denote the sum in parentheses as the usual sum of real numbers residing in Ajj; 77 the mark "-" signifies that the sum is undefined. Note, to conserve space, the table is completed by reflection across the diagonal. + re Ao re a;. reAi. reA^ reAf »€ Ao (r + «)o ir + sy„ ir + s)l„ ir + sfi ir + s)^ s&K (r + s); (r + «)A (r + 3)i if fc < n; else - (r + s)f if J < n; else - »eAi. (r + 3)j„ (r + a)l if A; < m; else - (r + s)f if J < m; else - sG A^ '(r + s)l - seA? (r + »)f Let n = F{A,R) and I>„ = Ao U (U^>„A;„); let Co = Ao U (IIn6u,A;,) and C„ = Pn II A^. Then the minimal prime ideals of H correspond to these maximal chains and are given by Q„ = {u G 71 : u{Cn) = {0}}, where n = 0,1,2, — Any supreraum \/j^jQj over a finite set J C a; is the prime ideal P^ given by {u € 71 : u(I>m) = {0}}) where m is the maximum element of J. Hence, for all new, the maximeil ideal Vo is not n-limbed since it is not a finite supremum of minimal prime ideals. However, Vq = Vnew^"' ^^^ ^ ^* ^^ ramified. D We show in Proposition 2.3.4 that for any nonisolated point p of a first countable space, the maximal ideal Mp is n-limbed for every n. If the space is also cozero-complemented, then rk(Mp) > 2* and Mp is 2*-limbed. Prom this, we also obtain the following, which is weaker than Theorem 4.1.2: Corollary 4.1.5. Every metric space is ramified. The following theorem means that if X is a metric space that is not pseudo- compact, then there exist points of ffX \ X such that Af is minimal. Hence, not every maximal ideal ofC{X) branches nontrivially in the root system Spec(C(X)). Recall that we call a topological space X perfect if every closed set of X is a G^-set. Note that any metric space is perfect. A point of PX is remote if it is not in the /?X-closure of any nowhere dense subset of X. A point p 6 I3X \ A^ is a C-point if p ^ intpx\x{dffxZ{f) r]pX\X) for all / e C{X). A theorem similar to the following appears in [W, 4.40]. All the proofs there carry through here, verbatim, under our reduced hypotheses. 78 Theorem 4.1.6. Let X be a completely regular space and consider the following conditions on a point p G PX \X: 1. p is a C -point. 2. Z[M^] contains no nowhere dense set. S. MP = OP. 4. p is a remote point. Without additional assumptions, (3) =r' (1). Let X be perfect and assume the exis- tence of a remote point, then (4) => (2) => (3). If X is perfect and the set of isolated points of X has compact closure in X, then (2) =» (4). If X is realcompact and C -points exist, then (1) =^ (2). It is not known if a remote point p always has the property that M^ = O^. We do know the following about the rank of a remote point: Proposition 4.1.7. Let X be a completely regular space. Let p € pX be a remote point. Then rkc(x){MP) = 1. Proof: In [vD, 5.2], it is demonstrated that no remote point is in the closure of two disjoint open sets of pX. Thus, in particular, no remote point is in the closure of two disjoint cozero-sets of pX. By [HLMW, 3.1], we have 1 = rk^xip) = 'rkc{fix){M*''). Finally, since C{pX) is rigid in C{X), we have rkc(x){MP) = 1. ■ Finally, we ask: does ramification of a point in X indicate ramification in PX, or vice versa? Proposition 4.1.8. A point p £ X is ramified in X if and only if it is ramified in PX. Likewise, a point p € vX is ramified in vX if and only if it is ramified in pX. Proof: This is a corollary of Proposition 2.2.12, since we know C{PX) = C*{X) 79 4.2 Ramified G^-points The main theorem, Theorem 4.2.5, of this section provides a good method for checking the ramification of G^-points. We will use it to characterize ramified Gtf-points in normal countably tight spaces and to find some ramified points in product spaces. We first discuss the following proposition, which we will obtain as a corollary to Theorem 4.2.5. Proposition 4.2.1. Let p € X be a Gs-point. If X\p is not C* -embedded in X, then rkip) > 2. Since an F-point has rank 1, the preceding proposition, proved in [Le] and (in greater generality) by van Douwen in [vD], shows that a G^-point, p, is not an F-point if it has the property that X \ p is not G*-embedded in X. We give a counterexample for the converse if the G^-condition is lifted. Example 4.2.2. Let X = BaCN, /3N \ N) be as defined in Example 5.2.5. There, we show that there exists a point p of the corona such that rk{p) = 2, X \p is G*-embedded in X and p is not ramified. But no point of the corona is a Gg. □ The following two results are Theorems 2.1 and 2.2 of [K2]; we will use these to prove our theorem on the ramification of G^-points. Theorem 4.2.3. Let p be a nonisolated Gs-point of X. If Z € Z[C{X \ p)] then dx{z) E z[c{x)]. ■" ' ■; " ^ "^ ' :■; ■■ ■\'. » ,. i : . > . ; i » ■ :• . „'", Define 7 : Z[C{X\p)] -^ Z[C{X)] by j{Z) = clx{Z). Let (^ be the extension of the identity map X \p — y X to the largest subspace Xi C p{X \p) such that it is extendible as a continuous map into X. 80 Theorem 4.2.4. Letp be a nonisolated Gs-point of X then 1. The mapping 7 is one-to-one from the set of prime z-fUters on X\p converging to points o/(A~n{p}) onfo the set of prime z-fUters on X contained properly inZ[Mp]. 2. A prime z- filter W on X\p converging to a point of(f)~^{{p}) is a z-ultrafUter if and only if j{W) is maximal in the class of prime z-filters on X contained properly in Z[Mp]. ,.^ Theorem 4.2.5. Let p be a nonisolated Gf-point of X. The point p is ramified if and only if X \p is not C* -embedded in X. Proof: Let p be a nonisolated G^-point of X. If p is not ramified then the prime z-ideal P = Y!,QeUm(M ) Q ^^ properly contained in Mp. We will show, in this case, that every point of X is the limit of a unique z-ultrafilter on X\p. Then by [GJ, 6.4] ,X\pis C*-embedded in X. Let q € X \ p. Then M, € Max(C(X \ p)) gives rise to the z-ultrafilter Uq — Z[Mg] on X \p. Clearly q e n{clx{U) : U G Ug}. The uniqueness of W, is a standard result [GJ, 3.18]. By [GJ, 6.3(b)], there exists a z-ultrafilter If on X\p converging to p. Assume that there exists another such z-ultrafilter, V. Let U = jU and V = 7V. Then Qu = Z^U and Qv = Zj[V are prime z-ideals of C(X) which are properly contained in Mp. liQu^P then U = Zx[Qu] § Zx[P]. Hence U = j'^U g ^-^Zx[P], which contradicts that W is a z-ultrafilter on X\p. Likewise, Qv is not properly contained in P. Thus P C Qu, P CQv and by [GJ, 14.8(a)], we must have either Qu C Qv or Qv C Qu- But Qu C Qv gives that W C V and therefore U QV. Since W is an ultrafilter, W = V, as desired. In a similar manner, if Qy C Qu, then V = U. Conversely, assume that p is ramified. Then Mp = Yj Min(Mp) and there exists more than one prime z-ideal in C(X) which is maximal in the class of prime 81 z-ideals properly contained in Mp. These give distinct prime z-filters on X which axe maximal in the class of prime z-filters on X properly contained in Z[Mp]. Hence, via 7, we have distinct ultrafilters on X \p converging to p. Again by [GJ, 6.4], X\pis not C*-embedded in X. ■ Corollary 4.2.6. If X is a metric space, then X is ramified. Corollary 4.2.7. If X is first countable and p €^ X is nonisolated, then X \p is not C* -embedded in X. Corollary 4.2.8. Ifp € X is a Gs-point and X\p is not C* -embedded in X, then rk{p) > 2. Let X and Y be completely regular spaces which are not P-spaces and let W = X xY. We conjecture that every nonisolated point of W is ramified. We use Theorem 4.2.5 to deduce two partial answers to this question. Proposition 4.2.9. Let X and Y be completely regular spaces and letW = X xY. Let X e X and y e Y be nonisolated Gs-points and let p = (x, y) € W. Then W\{p} is not C* -embedded in W. Proof: Since X is completely regular and z is a Gj-point of X, {x} is a zero-set of X. Say, {x} = Zxif) for some / e C{X). Then we have for E^ = {x} x Y, E. = {x}xY = n^{Zxif)) = Zw{fonx) where ttx denotes the natural projection from W onto X. Likewise, we have that Ey = X X {y} = Zw{g o TTy) where g e C{Y) such that Zy(g) = {y} and Try is the natural projection from W onto Y. Let / and 5^ denote the restrictions of forcx and gony to W\{p}. Then we have that E^\{p} = Zw\{p}{f) and ^„\M = Zw\ip}ig) are disjoint zero-sets of W\{p}. Thus Ex\{p} and Ey\{p} are completely separated in W\{p}. But p € clwiE^\{p})ndw{Ey\{p}) and therefore ^x\M and Ey\{p} 82 axe not completely separated in W. By the Urysohn Extension Theorem, W \ {p} is not C*-embedded in W^. ■ Corollary 4.2.10. Let X and Y be completely regular spaces and letW = XxY. Let X e X and y eV be nonisolated Gs-points and let p = (x, y) €. W. Then p is ramified inW. Proof: Follows from Theorem 4.2.5. ■ Proposition 4.2.11. Let X and Y be completely regular spaces and letW = XxY. Letp= {x,y) €W be nonisolated. IfW\ {p} is normal, then W\{p} is not C- embedded in W. Proof: Let Ex = {x}xY and Ey = Xx{y}. Then E^Xip} and Ey\{p} are disjoint closed sets in the normal space W \ {p}. Thus, they are completely separated in W \ {p}. But p e dw{Ex \ {p}) n clw{Ey \ {p}), so E^ \ {p} and Ey \ {p} are not completely separated in W. Therefore, by the Urysohn Extension Theorem, W\{p} is not C*-embedded in W. ■ Corollary 4.2.12. Let X and Y be completely regular spaces and letW = X xY. Let p= {x,y) eW be a nonisolated Gs-point of W. IfW\ {p} is normal, then p is ramified in W. Proof: Follows from Theorem 4.2.5. ■ We now investigate the ramification of G^-points in a class of spaces more general than metric or first countable spaces. ^.^ A topological space X is countably tight if for a subset U C X we have that any p 6 d{U) is in the closure of a countable set S CU. A Frechet- Urysohn space is one in which every p e cl{U) is the limit of a sequence of distinct points {pn} C U. It is evident that any Frechet- Urysohn space is countably tight. 83 Lemma 4.2.13. Let X be a noTTnal topological space and letp€ X be nonisolated. Then X\{p} is C* -embedded in X if and only ifp^ dx{A) n dx{B), whenever A and B are completely separated in X \ {p} . Proof: By the Urysohn Extension Theorem, X \ {p} is C*-embedded in X if and only if A and B are completely separated in X, whenever A and B are completely separated in X \ {p}. Assume that X \ {p] is C*-embedded in X and let A and B be completely separated in X \ {p}. Then A and B are completely separated in X and hence are contained in disjoint closed sets of X. Thus p ^ dx{A) H dx{B). Conversely, let A and B be completely separated in X\{p}. We wish to show that A and B are completely separated in X. By hypothesis, p ^ dx{A) n dx{B). Thus, dxiA) and dx{B) are disjoint closed sets of the normal space X. Hence, A and B are completely separated in X. ■ Proposition 4.2.14. If X is a normal countably tight topological space andp £ X is nonisolated, then X \ {p} is C* -embedded in X if and only if for every two countable sets 5i and S2 which are completely separated in X \ {p}, we have that p^clx{Si)ndxiS2). Proof: Assume that X \ {p} is C*-embedded in X and let Si and 5^2 be two count- ably infinite sets which are completely separated in X\{p}. Then by Lemma 4.2.13, we have that p ^ dx{Si) dx{S2)- Conversely, let A and B be completely separated in X \ {p}. Assume that p € dx{A) n dx{B). Then there exist countable sets Si C A and S2 C B such that p € dxiSi) n dx{S2)- Since A and B are completely separated, so are Si and 52. This contradicts the hypothesis. Thus by Lemma 4.2.13, A and B are completely separated in X and X \ {p} is C*-embedded in X. ■ 84 Corollary 4.2.15. Let X be normal Frechet-Urysohn space and letp £ X be non- isolated. Then X\{p} is C* -embedded in X if and only if there do not exist two sequences in X which are completely separated in X\ {p} and converge to p. Finally, Proposition 4.2.14 and Theorem 4.2.5 imply the following. Corollary 4.2.16. If X is a normal countably tight topological space andp G X is a nonisolated Gs-point, then p is ramified in X if and only if for every two countable sets Si and S2 which are completely separated in X \ {p}, p ^ clx{Si) O dx{S2)- 4.3 Ramification via C-Embedded Subspaces Let A and B be commutative rings with identity. Assume that we have a surjective ring homomorphism f : A-^ B, with K = Ker{f). Recall that there is a one-to-one order-preserving correspondence between Spec(-B) and the set of prime ideals P e Spec{A) such that K C P. Let M e Max(B) be such that M = Fi + F2 for some nonmaximal proper primes Pi, P2 ^ Spec (5). If AT g Spec (A) corresponds to M, then we have B/M ^ {A/K)/{N/K) ^ A/N. So N e Max(A). Let QuQ2 be the prime ideals of A corresponding to Pi and P2. Then (Qi + Q2)/K = Pi -|- F2 via the surjective map given by a-\-b i-^ f{a)-\-f{b) with kernel K. Thus, by the correspondence, we have that N/K ^ M = P1-I-P2 = {Qi+Q2)/K and TV = Qi-^Q2. In fact, we have shown: ] Proposition 4.3.1. Let A and B be commutative rings with identity such that there exists a surjective ring homomorphism f : A—^ B with K — Ker{f). If P < B is a prime ideal which is a sum of two distinct prime ideals then there exists Q 6 Spec (A) such that Q/K = P and Q is a sum of two distinct prime ideals of A. Ramification in a C-embedded subspace implies global ramification. Corollary 4.3.2. Let X be a completely regular space and let Y be a C-embedded subspace. If a point p ofYis 2-limbed in Y, thenp is 2-limbed in X. 85 Proof: Since Y is C-embedded in X, we have a surjective ring homomorphism from C{X) onto C{Y), given by restriction with kernel {/ G C{X) : Y C Zx{f)}- Hence this result follows from Proposition 4.3.1. ■ Proposition 4.3.1 also gives the following, since any compact subspace of a completely regular space is C-embedded. Corollary 4.3.3. 1. Let X be compact, Y C X a closed subspace. If a point of Y is 2-limbed in Y then it is 2-limbed in X. 2. Let X be a compact space consisting of more than one point. If every noniso- lated point in a proper zeroset of X is 2-limbed, then every nonisolated point of X is 2-limbed. Example 4.3.4. The Cantor Set is metric, hence every point is 2-limbed. By [E, 3.12.12c], every point of 2"^, the Cantor space of weight r, is contained in a closed set which is homeomorphic to the Cantor Set. Thus every nonisolated point of 2^ is 2-limbed. In fact, induction on Proposition 4.3.1 gives that every point of the Cantor space is n-limbed for every n € N. D Conversely, if a maximal ideal of A containing AT is a sum of primes containing K, then by the correspondence given above, its image is a maximal ideal which is a sum of primes in B. That is, if iV G Max(>l), K C N, ajid N = Qi-\-Q2 such that KCQiajudKC Q^, then f{N) e Max(B) as B/f{N) ^ {A/K)/{N/K) ^ A/N. And /(AT) = /(Qi + Q2) = /(Qi) + /(Q2) ^ Qi/K -\- Q^/K. This gives a partial converse: Proposition 4.3.5. Let A and B be commutative rings with identity such that there exists a surjective ring homomorphism f : A -¥ B with K = Ker{f). Let P be a prime ideal of A containing K. Then P is a sum of two distinct prime ideals 86 containing K if and only if f{P)/K is a prime ideal which is a sum of two distinct prime ideais of B. Corollary 4.3.6. The ring-homomorphic image of a commutative ramified ring with identity is ramified. Corollary 4.3.7. Let Y be a C-embedded subspace of X and let p eY have finite rank in X.Ifpis ramified in Y, then p is ramified in X. If any function in C{X) vanishing on Y also vanishes on a neighborhood of p, then p is ramified in Y if and only if p is ramified in X. Proof: The first statement is an application of Proposition 4.3.1 by induction. The second statement follows from Proposition 4.3.5 by induction. ■ Note that the hypothesis of the second statement of the above merely de- mands that the kernel of the restriction map be contained in O^ . This is satisfied if Y is open or if F is a P-set. The preceding results indicate that ramification is a local property. Let A be a commutative ring with identity, let 5 be a multiplicative system in A such that 1 G 5. Then there exists a one-to-one correspondence from Spec(5'~^>l) onto {P € Spec(>l) : POS = 0}. The proofs of the following are routine: Proposition 4.3.8. Let A be a commutative ring with identity, let S be a multi- plicative system in A such that 1 G 5. 1. IfP€ Spec(5'~M) is a sum of two proper primes, then the preimage ofP, the set {x € A : x/1 E A}, is a prime ideal which is a sum of two proper primes , in A. 2. If Q E Spec (A) is a sum of primes and Qd S = 0, then S~^Q is a sum of primes in S~^A. 87 Corollary 4.3.9. Let Y be a subspace of X such that C{X) ^ C{Y) is a ring of fractions map. That is, there is a multiplicative system S C C{X) such that 1 G 5 and C{Y) = S-'{C{X)). Then: 1. Forp G Y, rkvip) < rkxip)- 2. Let peYIfM}nS = 0, then there is P e Spec(C(X)) such that P^M} and rkvip) =rkx{P)- » ; CHAPTER 5 m-QUASINORMAL /-RINGS In [Lai -3], Suzanne Larson defines the notion of a quasinormal f -ring; one in which the sum of any two distinct minimal prime ideals is a maximal ^-ideal or the entire /-ring. We generalize this definition and a few of her results. 5.1 Definitions Definition 5.1.1. Let A be a commutative f-ring ivith identity and let m be a positive integer. Call A m-quasinormal if the sum of any m distinct minimal prime ideals is a maximal i-ideal or the entire f-ring A. If X is a topological space such that C{X) is m-quasinormal, then we call X an F^-space. Note that the "2-quasinormaJ" is Larson's "quasinormal" condition, the "1- quasinormal" condition is equivalent to von Neumann regularity, and if A is m- quasinormal then A is n-quasinormal for any n > m. Hence, the Fj-spaces are exactly the F-spaces and any F^-space is an F„-space, when n> m. Theorem 5.1.5 generalizes [Lai, 3.3] and characterizes the rri-quasinormal semiprime /-rings. Note that [Lai, 2.2], which we now state, gives necessary and sufficient conditions for a semiprime /-ring to have the property that the sum of any two distinct minimal prime ideals is a prime ^-ideal. This condition is stronger than the assumption we make in our theorem, but this result indicates when one can expect to be able to apply it. Theorem 5.1.2. Let A be a semiprime f-ring. The following are equivalent: 1. The sum of any two semiprime £-ideals is semiprime. 88 89 2. The sum of any two minimal prime £-id&ils is prime. 3. The sum of any two prime £-ideals is prime. 4- For any a,b e A^, the £-ideal a-'-'- + ft-*-^ is semiprime. 5. For any two disjoint elements a,fe e A+, the i-ideal a^ •{■h^ is semiprime. 6. For any a, 6 e A"*", the i-ideal a^ + b^ is semiprime. 7. When x, a, &, c, rf 6 -A+, a, & 7^ are such that x^ = c + d and aAc = bAd = 0, there exist g,h€i A such that x = g + h and gAa = hAb = Q. The theorem above holds for C(X) since the sum of two prime ideals is prime by [GJ, 14B]. We will use the following lemmas. The first follows from the fact that a prime ^-ideal P of a semiprime /-ring is minimal if and only if for every p ^ P there exists q ^ P such that pq = 0. The second lemma shows the existence of certain functions which we will take for grafted in the proofe to follow. Lemma 5.1.3. Let A be a semiprime f-ring in which the sum of any m distinct minimal prime ideals is a prime £-ideal. Let P be a prime i-ideal. Then P is minimal with respect to containing ]C^i ^f */ °"^ ^^^J/ */ f^^ every p £ P there exists q ^ P such that pq G YlJ=i ^f- Lemma 5.1.4. Let A be a commutative f-ring. Let n > 2 and let Qi,...,Qn be distinct minimal prime ideals. Then there exists an element f 6 Qj*" \ [Jj^Qj, for eachi = l,...,n. Proof: Let fk € Qt \Qk for A; = l,...,n. Then, by convexity, we have that / = VL2/ieQMUi^i<5i, as desired. ■ Theorem 5.1.5. Let m be a positive integer. Let A be a commutative semiprime f-ring with identity in which the sum of any m distinct minimal prime ideals is a prime i-ideal. The following are equivalent: 90 1. A is m-quasinormal. 2. For every nonmaximal prime £-ideal P, rk(F) < m — 1. 3. Let {aj}^i be a family of positive pairtvise disjoint elements of A. Proper prime i-ideals containing Y^=i ^f ^''^ maximal i-ideals. 4- Let {aj}p.i be a family of positive pairwise disjoint elements of A, let M be a maximal i-ideal containing the i-ideal ^17=1 °'fi '^^^ ^^* p € M. Then there exists z ^ M such that zp G Y^JLi ^f- Proof: (1) =>• (2) : Let P be a nonmaximal prime ^-ideal of A such that rk(P) > m and let Qi, . . . , Qm be m distinct minimal prime ideals that are contained in P. Then YlT-i Qj ^ ^ is not maximal, hence A is not m-quasinormal. (2) =» (3) : Let P be a prime ^-ideal containing Ylf-i af. Then afCP for every j. Therefore, Cj ^ 0{P) for all j, and hence P contains at least m minimal prime ideals by the pigeonhole principle. Thus condition (2) gives us that P is a maximal ^-ideal. (3) <=^ (4) : Follows from Lemma 5.1.3. (3) ^ (1) : Let M be a maximal ^-ideal and let Qi, • • ■ , Qm C M be minimal prime ideals. Then by hypothesis, Y^JjLiQj is a prime ^-ideal and we are left to show that it is a maximal ^-ideal. For each j = 1,.. .m, let Gj E Q'^\ lii^jQi and define hj = Aj^ijOj — AJt^iCife G Y^=i Qj- Then {6j}^i is a pairwise disjoint set of m distinct elements of Ylf^i Qj and by the choice of the Oj's, we have that AjfeLiO/t ^ ^T=iQk by convexity and Aj^jOj ^ Qj for each j. Hence bj ^ Qj and bf C Qj, for each j. Thus X)^i Qj is a maximal ^-ideal by condition (3), since E^i bf C ^^^ Qj. m The quasinormal condition is a variation of the normal condition, which is that the sum of any two minimal prime ideals of a semiprime /-ring with identity is the entire /-ring. This is discussed in [Hu]. The expected generalized definition 91 follows, along with a theorem recording two equivalent conditions. The result is a special case of Theorem 5.1.5 and the proof follows immediately from the one above and from [Hu, Theorem 8]. Definition 5.1.6. Letm>2 be a positive integer. An f-ring A is m-normal if for any pairwise disjoint family {aj}!^i, we have that A = Ylj=i ^j ■ Theorem 5.1.7. Let A be a commutative semiprime f-ring with identity and let m>2 be a positive integer. The following are equivalent: 1. A is m-normal. * 2. For any maximal t-ideal M, we have that rk(M) < m — 1. S. The sum of any m distinct minimal prime ideals is A. Before we move to describe F^-spaces, we first discuss a special class of Tn-quasinormal /-rings. Definition 5.1.8. Let A be a local f-ring. An embedded prime £-ideal P is high if for every minimal prime ideal N e Min(A), either NCPorN\/P = lex{A). Otherwise, P is low. Call an f-ring A a broom ring if for every maximal i-ideal M every prime i-ideal in Am is high. The following is immediate from Proposition 2.5.2 and Lemma 5.1.3. Proposition 5.1.9. Let A be a local commutative semiprime f-ring with identity and maximal i-ideal M. The following are equivalent: .• If r' .1 1. A is a broom ring. 2. T^{A) < 2. 3. If P C lex{A) is a prime i-ideal, then rk^(P) < 1. 92 4- If a,b G A are disjoint and P is a proper prime (.-ideal containing a^ + b^, then lex(A) C P C M. 5. Ifa,beA are disjoint and a-^ + b-^ C lex(A), then for every p G \ex.{A), there exists z ^ lex(A) such that zpea^ + b-^. Proof: (1) "^ (2) : Since every prime ^-ideal of A is high, we have that for any two distinct minimal prime ideals QuQ2 that Qi V Q2 = lex(A). Hence 7r(A) < 2 by definition. Conversely, Tr{A) < 2 implies that every minimal prime ideal is high and therefore that every prime ^-ideal is high, as desired. (2) •» (3) : Since 7r(yl) < 2, we have that rk(P) < 1 for any embedded prime ^-ideal P, by Proposition 2.5.2; and vice versa. (2) => (4) : By Proposition 2.5.2, we have that a-^ + b-^ = \ex{A). (4) <^ (5) : Follows from Lemma 5.1.3. (4) =^ (3) : Assume that (4) holds. Let P be an embedded prime ^-ideal and assume, by way of contradiction, that rk(F) > 2. Let QuQiQ P he minimal prime ideals. Let qi € Qt \ Q2 and let q2 ^ QtXQi- Disjointify by defining q] = Qj - Qi /\ 92 for j = 1,2. Then q^-^ C Q2 since q{ ^ Q2 and ^^ C Qi since qi ^ Qi- Then q{^ + ^-^ C P and hence lex (A) C P, which contradicts that P is embedded. ■ Ex£iinple 5.1.10. We now present an example of an /-ring which is 3-quasinormal but is not a broom ring. Let Aq = Ai = [0, 00) C R, and A2 = A3 = (1, 00) C R, where each is inversely ordered. Identifying the copies of in the disjoint union A^ = (Aq U Ai)/(Oo ~ Oi) and letting A^ be the disjoint union A2 U A3, we obtain a root system A = A^ U A^ with the induced ordering which we now describe. That is, r < s if and only if either r, s G Aj for j = 0, 1, 2 or 3 and r < s in the inversely ordered real numbers; or if r 6 A2, s 6 Ai; or if r 6 A3, s G Ai. Explicitly, r || s if r 6 A2 and s € Aq n A3 or if r 6 Ai and s e Aq. 93 We endow A with a partial binary operation. To begin, note that we define Oo + Oo = Oi + Oi = Oo + Oi = Oo '-- Oi. Let Tj, Sj £ A^ be nonzero for j = 0, 1, 2, 3. Let (*)j denote the sum in parentheses as the usual sum of real numbers residing in Aj] the mark "-" signifies that the sum is undefined. + ro so ri SI r2 S2 r3 S3 ro (2ro)o (ro + so)o - - - - — — so (ro + so)o (2so)o — - - - — — ri — (2ri)i (si +ri)i (r2 +ri)2 (S2+ri)2 (r3 + ri)3 (s3 + ri)3 SI _ - (ri+si)i (2s, )i (r2+Sl)2 (S2 + Sl)2 (r3 + Si)3 (S3 + Si)3 r2 _ — (ri + r2)2 (si +r2)2 (2r2)2 (S2+r2)2 - — S2 _ — (ri + S2)2 (si +82)2 (r2 + S2)2 (2S2)2 - - rs _ — (ri + r3)3 (si +r3)3 - - (2r3)3 (S3 + r3)3 S3 - - (ri + 83)3 (Sl + S3)3 - — (r3 + S3)3 (2S3)3 Let Co = Ao,Ci = Ai,C2 = Ai U A2, and C3 = Ai LI A3. Then the minimal prime ideals of F(A,R) are of the form Qj = {u e F(A,R) : supp(m) C A\Cj} for j = 0,2,3. The similarly defined Qi is a prime ideal by Proposition 3.3.15. Now, it is evident that L = lex(F(A, R)) = Qo V Q2 V Q3 and Q2 V Q3 = Qi / i- so we know that 7r(F(A,R)) = 3. Therefore, Proposition 5.1.9 shows that the ring is not a broom ring. Since L is the maximal ideal of the ring, we have shown that F(A, R) is 3-quasinormal. D We present a similar example of an broom ring that is not quasinormal. Example 5.1.11. Let Ao = [0,oo) C R, and Ai = A2 = (l,oo) C R, where each is inversely ordered. Let A^ = Aq and let A^ be the disjoint union Ai U A2 in order to obtain a root system A = A^ U A^ with the induced ordering which we now describe. That is, r < s if and only if either r, s e Aj for j = 0, 1 or 2 and r < s in the inversely ordered real numbers; or if r e Ai,s G Aq; or if r € A2,s € Aq. Explicitly, r || s if r € Ai and s e A2. We endow A with a partial binary operation. Let rj, Sj € Aj for j = 0, 1, 2. Let (*)j denote the sum in parentheses as the usual sum of real numbers residing in Aj-; the mark "— " signifies that the sum is undefined. 94 '0 ri_ «i «2 ro (2ro)o (rp + ao)o (ro + ri)i (rp-f8i)i (rp + rzja" {ro + S2)2 so l»P + rp)p (2ao)o (ao + ri)i (ap + ai)i (so + Ti)^' [SQ + S2)2 n (ri +ro)i (ri + ao)i (ri + 8i)i »l (ai + rp)! (ai +ao)i (ai +ri)i 1270T r2 (r2 +rp)2 (rz +ap)2 12^2)2 (r2 + 82)2 a2 (a2 + ro)2 (aa +ao)2 (a2 +r2)2 (2S2)2 Let Co = Ao,Ci = Ao U Ai and C2 = Ao U A2. Then the minimaJ prime ideals of F(A,R) are of the form Qj = {u e F(A,R) : supp(«) C A \ Cj} for j = 1,2; the ideal Qo, is a prime ideal by Proposition 3.3.15. Now, Qo = lex{F(A, R)) = Qi V Q2 and so we know that 7r(F{A,R)) = 2. Therefore, Proposition 5.1.9 shows that the ring is a broom ring. Since Qo is not the maximal ideal of the ring, we have shown that F(A, R) is not quasinormaJ. D 5.2 (B, m)-Boundarv Conditions Definition 5.2.1. Letm be an integer greater than 1 and let {Uj}]^^ be a family of m pairwise disjoint cozerosets of a topological space X. The subspace C(^=idxiUj) is called an m-boundary in X. Let B be a topological property. We say that a space X satisfies the (B, m)-boundary condition if every m-boundary in X has property 3. , ^ . , - In [Lai, 3.5], Larson proves that ii X is completely regular, every point of X is a Grpoint, and C{X) is quasinormaJ, then X satisfies the (discrete, 2)-boundary condition. This result is improved in [La3, 3.5] to say that if X is normal and for every p 6 0X \ vX, the ^-ideal O"' is prime then C{X) is quasinormal if and only if X has the (finite, 2)-boundary condition. Here we refine this theorem by removing the restriction on the points of the corona. First, a lemma, extending [La3, 3.1], which we henceforth refer to as "Larson's Lemma" : Lemma 5.2.2. Let X be normal and let {j9j}!^i Q C(X)'^ be a family of pairwise disjoint functions. Define Y^ = 07=1 dx{coz{gj)). Then Y^^Li of = Ciy^y^ My. 95 Proof: Each function in Y!^^i of must vanish on F^, hence YjJ^x df ^ (^y^Vm ^v For the reverse inclusion, we use recursion. The proof of the base case of m = 2 is in [La3, 3.1]. Let / G riyey^ ^y ^^ define /•/\^//^l ifxecZ(co2;(5i)), Since X is normal and f\ is defined on a closed set, the function has a continuous extension, /i € C{X). Then / A 1 -/i 6 5i^ and 7i e fK^^p = P e 0^2 c/(co2:(5j))}. Recursively define a function /n. to be the continuous extension of ^*^"^ - \0 if X £ d{coz{gk)) xer\]lk^,cl{coz{gj)). Then 7fc_i A 1 - 7fc G ^j^ and 7fe G fli^P = P ^ fl^it+i c^(«>^(5i))}- In particulai, by the base case, we have that 7m-2 € C\{^p '■ P ^ PijLm-i ^i<^^i9j))} = 9m-i + 9m- But then 7m-3 ^^ 1 - 7m-2 e ffm-2 implies that 7m-3 ^ 1 G YlT=m-2 9f and therefore we have that 7m-3 = (7m-3 A l)(7m-3 V 1) G YJj=m-29f- Thus, by recursion, we deduce that 7i G E^2^/- Hence /Al = (/Al-7i)+7i ^ E^i5i^ and therefore, /=(/ A l)(/Vl)GEr=i^^ as desired. ■ Let X be normal and let K^, as above, be given. The set Y^ is C-embedded in X by [GJ, 3D]. Thus we have a surjective ring homomorphism ip : C{X) -» C(y^) given by restriction of functions. The kernel of the homomorphism is 171 K = {feC{X):Yr.CZx{f)}= n Mv = Y.sf^ by Larson's Lemma. Thus by the First Isomorphism Theorem, it follows that m C{Y^)<^C{X)/K = C{X)lY,9f- We utilize the one-to-one correspondence between the prime ideals of CiXrr^ and prime ideals of C{X) which contain K, the kernel of v- 96 Theorem 5.2.3. Let X be normal and m > 2 an integer. The ring C{X) is m- quasinormal if and only if X satisfies the {P,m) -boundary condition. Proof: Let Y^ = fl^i cl{coz{9j)) be an m-boundary. Y^ is a P-space if and only if the prime ideals of C{Ym) are both minimal and maximal by [GJ, 4J]. By the discussion above and Larson's Lemma, this is also equivalent to the condition that the prime ideals of C{X) containing YJ^liOf ^^ maximal. In turn, this is equivalent, by Theorem 5.1.5, to the statement that C{X) is m-quasinormal. ■ By [GJ, 4K.1], we have that countably compact F-spaces must be finite and discrete. In this light, the following is immediate. Corollary 5.2.4. Let X be normal. t-.; :• ... 1. If X is countably compact, then C{X) is m-quasinormcU if and only if X satisfies the {&nite,Tn) -boundary condition. 2. If X is locally compact, then C{X) is m-quasinormal if and only if X satisfies the {discrete, m) -boundary condition. 3. If X is a-compact, then C{X) is m-quasinormal if and only if X satisfies the (countable discrete, 7n)-6o«nrfan/ condition. Example 5.2.5. (Butterfly Spaces) Let X be a noncompact completely regular space and let S C px \X he closed. Define m Brr.{X,S) = (]j0Xj)/iS,r^S2 Sm), J=l where we assume that Xj = X,Sj = S for all j = 1, . . . m and Si ~ Sj indicates that corresponding points of S are identified, when i ^ j. Consider X = B„(N, /?N \ N). Let gj{x) = ^ when x eNj and let gj vanish elsewhere on X. Then {gj}j^i is a pairwise disjoint set of functions such that we 97 have y„ = nr=i dx{coz{gj)) = /?N \ N, which not a P-space. Thus B^(N, /?N \ N) is not an F^-space by Theorem 5.2.3. Every point of this space has rank less than or equal to m, thus it is an F^+i-space, by Theorem 5.1.5. The minimal prime ideals contained in Mp for p € /9N \ N and j = 1,... ,m, are given by: Qj = {/ e C(B^(N,/3N \ N)) : p 6 m«^N,ci^N,^N, (/In,)}. We show that C(B,„(N, /?N\N)) is a broom ring by demonstrating that Qj, C Qj+Qj for all j > 3. Let f ^ Q^ and represent the function by an m-tuple (/i, /z, • • . , fm) where fk = /I/jn* for all A; = 1, . . . , m. For some j > 3, let 9i = {fjj2,f3,---,fm) and 92 = {fi- fj, 0,0,..., 0). Then ft e Qj, for i = 1, 2 and / = 51 + 52 G Qp + Qp- Thus we have shown that 7r(C(B^(N, (m \ N))mp) < 2 for all p G B^(N, /?N \ N). Hence C(B„(N, /?N \ N)) is a broom ring by Proposition 5.1.9. Let (f : C{Bm{^, /3f^ \ N)) -> C{Ym) be the canonical surjection with kernel K = YlT-idf' ^® want to explicitly describe the root system structure at each point of the space 5m(N, /?N \ N). For each j, the points p 6 Nj are isolated and hence the spectrum at each of these points consists only of the maximal ideal. If p e ^N\N is a P-point then Mp E Max{C{Ym)) is minimal. Hence Mp = (p^{Mp) is a maximal ideal of C(J3m(N, /3N \ N)) and is minimal with respect to containing K. Thus, the maximal ideal Mp is ramified and the graph of Spec(C(PTO(N, /?N\ N)jg ) has the form: (5.1) 98 Ifpe/9N\Nisan F-point which is not a P-point, then Mp € Max(C(Fm)) properly contains a unique minimal prime ideal, namely Op. Hence Mp = (p^{Mp) is a maximal ideal of the ring C{Bm{N, pN \ N)) and the ideal Op = (f'^iOp) is the unique prime ideal of C(B^(N, /?N\N)) which is minimal with respect to containing K. Thus, Ql+Ql = DJLi Q^ = Op S Mp and the graph of Spec(C'(5„(N, /5N\N)j^J has the form: (5.2) The following is also immediate from Theorem 5.2.3: Corollary 5.2.6. Let X he normal. 1. Let X he a noncompact, locally compact, a-compact F-space and S a closed suhspace of the corona. Then B2{X, S) is an F-space if and only if S is finite. 2. Let X he an F-space and S a nowhere dense zeroset of X. Let Xj = X and let S = Sj C Xj for j = 1,2. Define Y = {X^U X2)/{Si ~ S2), where corresponding points of S are identified. Then Y is an F^-space if and only if S is a P-space in the suhspace topology. Proof: (1) By [GJ, 6.9], the hypotheses on X imply that fiX \X is compact and A" is a cozeroset of PX. Thus S is compact and there exists / G C{I3X) such that coz{f) = X. Let ' f{x) if X e pXi ^ ,_, j f{x) if xe 13X2 /i(x) = t if X 6 X2 ^W = {r iix e Xi 99 Then d{coz{fi)) n d{coz{f2)) = S. If B2{X, S) is an Fa-space, then 5 is a compact P-space by Theorem 5.2.3. Thus, S is finite by [GJ, 4K.1]. Conversely, assume that S is finite. Since X is an F-space, we know that any 2-boundary of B2{X,S) must be contained in S and must therefore be finite. Thus, every 2-boundary is a P-space and we conclude that B2{X, S) is an Pa-space by Theorem 5.2.3. (2) As above, any 2-boundary of Y is contained in S. In particular, 5^ itself is a 2-boundary since it is a nowhere dense zeroset of X. Thus, if y is an Pa-space, then 5 is a P-space, by Theorem 5.2.3. Conversely, since any subspace of a P-space is a P-space by [GJ, 4K.4], we know that every 2-boundary of F is a P-space. Therefore, Y is an Pa-space. ■ Next, we show that it is the case that any normal P-space arises as a 2- boundary in an Pa-space. " "' ' ; " ;; j. •? i • ■ - -^. ... ■..- ■ .... Proposition 5.2.7. LeX Y he a normal P-space. Then there exists an F2-space X containing Y such that in the subspace topology Y = clx{coz{fe)) clx{coz{fo)), where fe,fo £ C{X)'^ are disjoint. Proof: Let F be a normal P-space and for each y G F, let ayNy be the one-point compactification of the natural numbers in which ay plays the role of the point at infinity. Define X = (Y U iUy^Y ^v^v)) / i^v ~ v)- ^^^ *^^ points of N remain isolated. A base for the ^-neighborhoods of y G F is given by the sets of the form ^y U (Uj/'eu ^y*)) where Uy is a neighborhood of y in F and Ny> is a neighborhood of ttj,' in OCy''Ny'. Define fe on X such that /e(n) = ^ for all even n £ Ny and fe{x) = otherwise. Similarly, define a function /„ on X such that foin) = - if n G Ny is odd, and foix) = otherwise. Then fe^fo = and F = dx{coz{fe)) n dx{coz{fo)). 100 Since the only points of X having rank greater than 1 are contained in the subspace F, we have that all the 2-boundaries are subspaces of a P-space. Thus by [GJ, 4K.4] and Theorem 5.2.3, we have that X is an F2-space. ■ It is natural to consider the implications of boundary conditions other than for F-spaces. We now take a look at F-space boundary conditions. Theorem 5.2.8. Let X be normal. The following are equivalent: 1. X satisfies the (F— space, n)-6o«ndarj/ condition. 2. Let p e fix. If {Pi, ... , P2n} ^ MP are pairwise incomparable prime ideals, then X)j=i Pj '^"'^ J2j=n+i Pj °^^ comparable. 3. There does not exist a pair of noncomparable prime ideals contained in the same maximal ideal such that each has rank greater than or equal to n. That is, the graph of Spec(C(X)) does not contain a copy of Proof: The equivalence of (2) and (3) is clear. (1) ^ (3) : Let p e 0X be such that M^ contains two incomparable prime ideals, F, Q, each of rank greater than or equal to n. Let F^, . . . , F„ C F and Qi, ■■■,QnQQ be distinct minimal prime ideals. For A; = 1, . . . , n, let /ifc e Qt \ {{Uj^kQj) u (Ui^kPi)) f2k e Pi: \ ii^j^kQj) u {\Ji^kPi)). Then by convexity and primeness, fk = /ifcA/afc G {Qtf^Pk)\iiUjjtkQ3)^{l}ijtkPi)) We disjointify by defining gk = Aj^fe/i - /^%ifj- Then gk ^ Qk^ Pk and hence, 9i c Q, n Pk. Thus ELi ^i^ ^ (ELi Qk) n (ELi Pk)QPn Q. 101 If y = Q"^^ d{coz{gk)), then by the prime correspondence discussed before Theorem 5.2.3, and by the construction of the functions {gkjk^^i, the incompa- rable prime ideals P and Q of C{X) give rise to the incomparable prime ideals PlYrk=i9i,QIY.l=i9i Q MVELi5i^ i« C{Y). Therefore, y is an n-boundary which is not an F-space. (3) =^ (1) : Let F = n"=i c/(co2;(/j) be an n-boundary in X and assume that Y is not an F-space. Then there exists p €. PV such that M^ contains incomparable prime ideals P, Q. Since C{Y) ^ C{X)/ Yl%i fj-, we have that M^ contains the two incomparable prime ideals P, Q such that P/ Yl"=i fj- = P and Q/ YTj=i // - Q- Thus, since YTj=i ft - ^' ^^ ^°°^ *^^* '"^c(x)(^) > ", by the pigeonhole principle and [BKW, 3.4.12]. Likewise, rkc{x){Q) >n. m Example 5.2.9. Consider the ordinal X = iJ^ -Vim the interval topology. Let E be the subset {A-|-2n:nGci;,A limit ordinal} be the set of even ordinals less than a;^ -I- 1 and let D be the subset {A -I- (2n -t- 1) : n € a;, A limit ordinal} be the odd ordinals less than w^ -(- 1. Define /e(x) = - if x = A -I- 2n £ F, and let /g vanish otherwise on X. Let /o(x) = Mf x = A-l- (2n-f-l) € D and let /„ vanish otherwise on X. Then Y = dx{coz{fe)) fl c/x(co2(/o)) = {no; : n e w} U {o;^} is homeomorphic to the one point compactification of u where u^ acts as the point at infinity. This 2-boundary is not an F-space. Thus, via analysis similar to the proofs above, the canonical surjection and [GJ, 14G], we know that the spectrum at A4,s in C{u^ + 1) contains the following subgraph: (5.3) 102 In fact, if, for m > 2, we partition N into m countably infinite sets {Aj}^^i and let . , . \ - if X = A + n for n e A j ^W=|o else. Then Y = fl^i cl{coz{fj)) = {nu : n G a;}U{a;^} is an m-boundary in w^ + l. Thus each branch in the graph above that emanates from the maximal ideal has at least 2' minimal vertices, by Proposition 2.3.3. D The special F-spaces, extremally and basically disconnected spaces, have a different sort of boundary description. Let us review a couple of results. Recall that a completely regular space X is cozero- complemented if for any cozeroset U C X there exists a cozeroset V C X such that U DV = and U UV is dense in X. Dually, an /-ring A is complemented if for every f G A^ there exists g E A'^ such that f Ag = Q and f + g is not a zero-divisor. Lemma 5.2.10. Let X be completely regular. 1. The following are equivalent: (a) C{X) is complemented, (h) X is cozero-complemented. (c) Min(C(X)) is compact. 2. X is a cozero-complemented F -space if and only if X is basically disconnected. 3. X is extremally disconnected if and only if Min{C{X)) = Max(C(X)) is ex- tremally disconnected. Proof: (1) The equivalence of (a) and (c) is shown in [M, 1.5]. The equivalence of (a) and (6) is from the definitions. (2) See [HVWl, 2.16]. (3) See [M, 2.6] and [M, 2.7]. ■ 103 Let F C C{X). Under the hull-kernel topology on Spec(C(X)), we define a subspace homeomorphic to Spec(C(X)/ Ylf^p /"^) by Vjr = {P e Spec(C(A")) : P is minimal with respect to containing ^ /"''}• Proposition 5.2.11. Let X he normal. The following are equivalent: 1. X soiis^fts i/ie (basically disconnected, n)-6oundan/ condition. 2. X satisfies the (cozero -complemented F, n)- boundary condition. 3. If F = {fj}]=i C C(X)+ is o pairwise disjoint family, then Vf is compact and there does not exist a pair of noncomparable prime ideals contained in the same maximal ideal such that each has rar^ greater than or equal to n. Proof: (1) ^ (2) : Follows from (2) of Lemma 5.2.10. (2) ^ (3) : The compactness of Vf follows by (1) of Lemma 5.2.10 and the prime correspondence discussed before Theorem 5.2.3. The final statement of (3) is a result of Theorem 5.2.8. ■ Proposition 5.2.12. Let X be normal. The following are equivalent: 1. X satisfies the (extremally disconnected, n) -boundary condition. 2. If F = {/j}"=i C C{X)'^ is a pairwise disjoint family, then Vf is extremally disconnected and compact and there does not exist a pair of noncomparable prime ideals contained in the same maximal ideal such that each has rank greater than or equal to n. Proof: Follows directly from (3) of Lemma 5.2.10, the prime correspondence dis- cussed before Theorem 5.2.3 and from Theorem 5.2.8. ■ Next we describe the spectra of C{X) which guarantee the (F^, n)-boundary condition on X. . \ 104 Definition 5.2.13. Ze* a € N and letn>0 be an integer. Then we call the n-tuple a = (ai,02,...,an) a good (ordered) paxtition of a if the following hold: for all j = l,...,n, we have that < %■ G Z, o = ai + 02 H f- On, ai > 1 and ci > 02. Construct an upwaxd-directed graph Aq corresponding to a good partition a of a containing: nodes Pi,P2,--,Pn such that Pi < P2 < • • • < Pn, each pj lies above 5j = tti + 02 H \-aj terminal vertices and there is a node q such that q > pj for every j = 1, 2, . . . , n. As follows: (5.4) Proposition 5.2.14. Let A be a commutative semiprime f-ring with identity in which the sum of any m distinct minimal prime ideals is a prime £-ideal. Then A is m-quasinormcd if and only if Spec^(>l) does not contain A^ as a subgraph for any good ordered partition aofm. Proof: Assume that Spec^(>l) contains A^ as a subgraph for some good ordered partition a = (oi, . . . , o„) of m. Let Pk be the prime ^-ideal at the node pk for each A; = 1, . . . , n. For a fixed k, let Qk = {Qkj}%i be a family of distinct minimal prime ideals contained in Pk but not contained in Pj for j < k. Then [J"^^ Qk is a family of m distinct minimal prime ideals whose sum is the nonmaximal prime ^-ideal P„. Thus A is not 7n-quasinormal. Conversely, if A is not m-quasinormal, then there exists a family of m distinct minimal prime ideals {Qk}^=i such that Y^^=i Qk is not a maximal ^-ideal. Let 105 Pi = Yjk=i Qk ^^ ^^* ^ ^^ *^^ maximal ^-ideal containing it. Then the subgraph of Spec (A) bounded by the vertices corresponding to the prime ^-ideals in the set {Pi,M} U {Qfc}JbLi is of the form A^ for some a. m Example 5.2.15. The good ordered partitions of 4 are (4), (3, 1), (2, 2) and (2, 1, 1). UX is normal, then it satisfies the {P, 4)-boundary condition if any only if Spec(C(A")) does not contain any subgraphs of the following forms: (5.5) This follows from Theorem 5.1.5 and Proposition 5.2.14.n Obtain the graph A* by appending a graph having at least A; > 2 minimal vertices to each terminal vertex of the graph A^. For instance: (5.6) The following is a consequence of Proposition 5.2.14 and the method of proof of Theorem 5.2.3. Corollary 5.2.16. Let X be normal. Then the following are equivalent: 1. X satisfies the (Fm, k) -boundary condition. 106 2. Spec(C7(A')) does not contain a subgraph of the form A* for any good ordered partition a of m. 3. Let a family V C Spec(C(X)) ofmk noncomparable prime ideals and a good partition a = (oi, 02, . . . , a„) ofmk be given. Partition V into pairwise disjoint sets {Vj}]^i where Vj contains aj elements ofV. Then at least two of the prime ideals given by ^ Vj are comparable. . Proof: The equivalence of (2) and (3) is clear. (1) => (2) : Assume that X contains a fc-boundary Y = ff^j d{coz{fi)) which is not an Fm-space. Then by Proposition 5.2.14, Spec(C7(y)) contains a subgraph of the form Aa below some maximal ideal My for some good partition a = (ai, . . . , a„) of m and some p E f3Y. For each i = 1, . . . , n, let Pj be the prime ideal at the node Pi G Aq and let {Qj^}^i be a family of distinct minimal prime ideals contained in Pi but not contained in Fj for / < i. Then, by the prime correspondence discussed before Theorem 5.2.3, there exist, for each i, prime ideals Pi,Qij in C{X), such that Pi ^ Pi/Eti fi^ and Qi^ ^ Qij/Eti ft for each 3. Then by [BKW, 3.4.12], rk(Qi ) > k for all i, j. Therefore, the spectrum below M^ contains a copy of A*. (2) =^ (1) : Assume that the spectrum below M^ contains a copy of A* for some good partition a = (oi, . . . , a„) of m. For each i = 1, . . . , n, let Fj be the prime ideal at the node Pi G A* and let {Qi }^i be a family of distinct nonmaximaJ as well as nonminimal incomparable prime ideals contained in Fj, but not contained in Fj, for / < i. For each pair i, j such that \ <i <n and 1 < j < Cj, let {Q'.}f=i be a family of distinct minimal prime ideals contained in Qi- but not contained in Qr, , if r < i or if r = i and s ^ j. Let fstr £ Q\t \ (U i¥^ Q\t)- Then, by convexity, we have that for all l<a<n l<t<a, ^ < r < k, fr = Mi<s<nfstr e {r[i<s<nQ\-t) \ (U H^r Q[X Disjointify thesc l<t<o, l<t<o, l<»<n l<t<a, functions via gr = A„^r/« - ^t=ifv t Q^at- Then g^ C Q^^ for all s,l Finally, 107 Er-i 9r ^ Qij for all pairs i,j. Then Y = ff^^ d{coz{gr)) is not an Fm-space since M^ contains A^ as a subgraph, by our choice of the functions. ■ 5.3 0X, m-Quasinormal and SV Conditions First we give equivalent conditions under which one may expect that /3X is an Fm-space for some integer m. This improves [La2,4.3] and extends [La2, 3.3]. Lemma 5.3.1. Let X be normal and let Y^^ {x e X : rkx(x) > m - 1}. 1. Let y^(3X\X. //rk^x(y) > m - 1, then y 6 dpx{Yra)- 2. IfYm is compact, then rk^xiy) <m-l for every y € PX \ X. Proof: (1) Let ?7 be a /?X-neighborhood of y. Then there exists a closed PX- neighborhood V of y such that V C.U and F n X is closed in X. Since V r\X is C*-embedded in X, we have that y e /?(y n X) = dpxiV r\X)CV. Now, by [La2, 1.6], we have that V n F^ y^ since rk^x(y) > m - 1. (2) Let y^(3X\X.Vi. rk^x(y) > m - 1, then by (1), y e d^xiX^) Q X, which is a contradiction. Hence, rk^x(F) < m — 1. ■ Using the lemma above and Theorem 5.2.3, we obtain: Theorem 5.3.2. Let X he normal and let m>2 be an integer. The following are equivalent: 1. C{X) is m-quasinormal and Tkx{M^) <m—l for every p E jSX \ X. 2. X satisfies the {'&nite,Tn) -boundary condition. 3. X contains only finitely many points of rank greater than m — 1. 4. PX contains only finitely many points of rank greater than m — 1. 5. C{/3X) is m-quasinormal. 108 Proof: The implications (2) =^ (3) =» (4) =^ (5) have the exact same proofs as in [La2, 4.3], using Lemma 5.3.1. (5) =^ (2) : Let {co2;(/j)}J^i be a pairwise disjoint set of cozerosets of X, and for each i, let /f be an extension of f, to PX so that {coz{fi)}iLi is a pairwise disjoint family. Then fllli dfix{coz{ff)) is finite by Theorem 5.2.3. Since fli^i dx{coz{fi)) is contained in this finite set, it too is finite. Thus, (2) holds by Theorem 5.2.3. (1) =j> (5) : Follows from Theorem 5.1.5. (3) ^ (1) : Since (3) implies (5), we have that C[X) is 77i-quasinormal. Condition (3) also implies that xk^xip) = 1 for all p € f5X \ X, by Lemma 5.3.1. ■ Recall that an /-ring A is called an SV-ring if A/P is a valuation ring for every prime ideal P. The following arises when in pursuit of conditions which imply that C{X) is both r/vquasinormal and SV. Theorem 5.3.3. LetX be normal andm > 2. IfC{X) is m-quasinormal and every maximal £ -ideal has finite rank, then X satisfies the {Qmte,m) -boundary condition. Proof: Assune that C{X) is m-quasinormal, that every maximal £-ideal has finite rank and that {Uj : I < j < m} is a family of pairwise disjoint cozerosets such that the set W = rij^idxiUj) is infinite. Then there exists a copy of N in W. Denote this copy by F = {xj : j € N}. Then Y is not closed in PX, so there exists p € /3X such that p e d^xiy) \ Y- Let {t/j : 1 < j < m} be cozerosets of PX such that U'jDX = Uj for each j. Then p ^ C/j for all j = 1, 2, ..., m and p e nf^id^xU'y For each i, Xi E W and hence rkx{xi) > m. Let Pii,P2t>"-j-fmt be distinct minimal prime ideals of C{X) such that we have Pji C M^^ for all j = 1,2, ...,m and Uj ^ Uife/jC02;(Fjfci). Let p 6 PX and rkc{x)iM^) = n < oo. Then as shown in the proof of [L,4.2], there exists a minimal prime ideal Q C Af such that for every Z e Z{Q) such that Z n F is infinite. 109 For every / G C{X) define JJ = {xi G Y : f e Pji} for j = 1, 2, ..., m. Let ^ be the 2-ultrafilter on Y containing the z-filter {Z HY : Z e Z{Q)}. Define for j = 1,2, ...,m, i2j = {f e C{X) : Jj e J^}. As shown in [L,4.2], each Rj is a prime ideal of C{X). First we show that the Rj are noncomparable ideals. For each Xj G F, let C/j be a neighborhood of Xj such that the collection of these neighborhoods is pairwise disjoint. For each i e N and for each 1 < /, fc < m, let fiki e C{X) such that fiki e Pii \ Pki, f{X \ Ui) = 0, and < /,« < ^. Then let /,* = YlZi hki- FinaUy, fik^RiX Rk and /« eRk\Ri for all 1 < i, A; < m, / ^ A:. Let /2j C /2j be a minimal prime ideal for each j = 1, 2, ..., m. Larson demon- strates that Rj C MP for all j. We will show that Yl'JLi ^j is neither the maximal £-ideal Mp nor all of C{X). Define h e C{dxY) by h{xi) = 4 for i = 1, 2, ... and let h{x) = otherwise. Then since X is normal and dxY is closed, h extends to a function h eC{X) such that h € M". We show that h i Xl^i -^i- For each j = 1, 2, ..., m, let /, e ilj. Then jj. 6 :F and hence f%iJ}. G :F and (X;7=i /j)(xi) = for every Xi G nf^^ J}.. If h = 5]^i /j then since h{xi) / for all Xj G F, = n^^ Jj. G :F, a contradiction. Thus /i G M" \ X:7^i ^i and therefore, E^^i ^j ^ Er=i Rj S ^^ ^^ ^(^) ^^ not T7>-quasinormal, a contradiction. Thus for any m pairwise disjoint cozerosets {Uj : 1 < j < m}, the set Hf^^dxUj is finite. ■ REFERENCES [AF] M. Anderson and T. Fell, Lattice- Ordered Groups, Reidel (Kluwer), Dor- drecht, 1988. [AM] M. Atiyah and I. 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Amer. Math. Soc. (41), 1937, pp. 375-481. [V] J. Vermeer, The smallest basically disconnected preimage of a space, Topol. and its Appl. (17), 1984, pp. 217-232. [Wa] R. Walker, The Stone- Cech Compactification, Ergebnisse der Mathematik und Ihrer Grenzgebiete, Springer Verlag, Berlin-Heidelberg-New York, 1974. [Wo] S. Woodward, On f -rings which are rich in idempotents. Doctoral Dis- sertation, University of Florida, 1992. BIOGRAPHICAL SKETCH Chawne Monique Kimber was bom in Frankfort, Kentucky, on January 12, 1971. She was raised in Tallahassee, Florida, where she graduated from Leon High School in 1988. Chawne received the Bachelor of Science in Mathematics from the University of Florida in 1992, and the Master of Science in Mathematics from the University of North Carolina at Chapel Hill in 1995. 113 I certify that I have read this study and that in ray opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of philosophy ^-^ " limk Jorge Martinez , Chairman Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ichard Crew Associate Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Alexander Drahishnikov Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Scott McCuUough Associate Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable stajidards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Mildred Hill-Lubin Associate Professor of English This dissertation was submitted to the Graduate Faculty of the Depaulment of Mathematics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 1999 Dean, Graduate School cKiHT'yA^'i Buoum^v'cx/ io BO^'TH ■■". rjf^'^oi siam LD 1780 199^ i<i]/A^i :i\^^^w;<jii ^v^fr^iy:) }■: V At, ' A? <! 10 a-:!i5JfO UNIVERSITY OF FLORIDA lliliiilli 3 1262 08554 8427 Aaiil0.n ^O YTIRflS vli'lu *i-]0 *